Logic and Cantor

[Brian]

Hello Everyone!

There are two parts to this post. The first part has to do with the debate between Modern and Traditional (Aristotitlian) Logicians concerning "existential import." Being a Christian, I believe that logic reflects the way God thinks. In regard to the debate, I believe that Modern Logic is more reflective of the way God thinks than Traditional Logic. Would anyone like to defend the Traditional position?

The second part of the post has to do with the Transfinite and God's omniscience. George Cantor in the 19th century paved the way for the mathematics of the 20th century with his proof concerning the uncountability of real numbers. This theorum is the only basis for all set theory and meta-mathematics concerning the Transfinite. Recent attacks on God’s omniscience employ a metaphysical application of Cantor’s theorem. The basic idea is that if R is uncountable, then God cannot be omniscient. Rather than attack the validity of the implication, I would like to discuss the validity of Cantor's proof. I claim that Cantor was wrong, and that all mathematics based on Transfinite numbers (which encompasses most of the mathematical breakthroughs of the 20th century) is built on a foundation of sand. I say the storm has come, and great was Cantor's fall! Therefore, the implication concerning God's omniscience is moot.

All of you philosophy buffs out there who believe R is uncountable, step up to the plate!

Sincerely,

Brian

[garthoverman]

Today @ 05:24 PM post located here (http://www.theologyweb.com/forum/showthread.php?s=&postid=132339#post132339)
Brian:

Hello Everyone!

There are two parts to this post. The first part has to do with the debate between Modern and Traditional (Aristotitlian) Logicians concerning "existential import." Being a Christian, I believe that logic reflects the way God thinks. In regard to the debate, I believe that Modern Logic is more reflective of the way God thinks than Traditional Logic. Would anyone like to defend the Traditional position?

The second part of the post has to do with the Transfinite and God's omniscience. George Cantor in the 19th century paved the way for the mathematics of the 20th century with his proof concerning the uncountability of real numbers. This theorum is the only basis for all set theory and meta-mathematics concerning the Transfinite. Recent attacks on God’s omniscience employ a metaphysical application of Cantor’s theorem. The basic idea is that if R is uncountable, then God cannot be omniscient. Rather than attack the validity of the implication, I would like to discuss the validity of Cantor's proof. I claim that Cantor was wrong, and that all mathematics based on Transfinite numbers (which encompasses most of the mathematical breakthroughs of the 20th century) is built on a foundation of sand. I say the storm has come, and great was Cantor's fall! Therefore, the implication concerning God's omniscience is moot.

Couple of things:

I've seen the proof for the uncountability of R, and as far as I can tell, if you deny its validity, then you must deny one of the fundamental axioms that defines R. I would be interested in hearing your objections to it.

Also, I fail to see how the one might mount an attack of God's omniscience from the premise of the uncountability of R. As it appears to me, if it is impossible to ennumerate R, then it is impossible to know every element of R. Where God's omniscience is defined as "knows all that is possibly knowable," and if the cardinality of R is unknowable in a real sense, then God's omniscience is still preserved--there just exist some things that are unknowable even for God.

I dunno... those are just my first gut reactions...

BTW - As at least one Tweb participant will tell you (IIRC his real-life work involves structural engineering) sand actually makes a rather reliable foundation... :teeth:

Yours,
Garth

[Rand]

Today @ 05:24 PM post located here (http://www.theologyweb.com/forum/showthread.php?s=&postid=132339#post132339)
Brian:

All of you philosophy buffs out there who believe R is uncountable, step up to the plate!


I am guessing that the proof you have seen for the uncountability of the continuum is a diagonalization argument where we assume that the real numbers are countable, list them, and then show a real number that is not in the list, thus proving by contradition that the real numbers are indeed uncountable. I am not sure what your objection might be to this proof-- perhaps you can tell us.

Of course, there are other ways to get there as well. First we would establish the theorem that says no set is equivalent to its power set (I trust you have no problem with this result). Then, we can show that the set of real numbers is equivalent to the power set of the natural numbers. Thus, the natural numbers are not equivalent to the real numbers (i.e. the set of real numbers is uncountable).

Furthermore, I do not see how someone could mount an attack on God by using Cantor's results. God, by definition, would be beyond such trivial things as human abstractions (like the real number system).

Well, I am leaving work now... I might go further into this later when I get home.

Rand

[Brian]

Hello Garthoverman!

Thank you for your first gut reactions.


I've seen the proof for the uncountability of R, and as far as I can tell, if you deny its validity, then you must deny one of the fundamental axioms that defines R. I would be interested in hearing your objections to it.

I do not believe that the fundamental axioms that define R are denied in the refutation of Cantor's proof. There are logical errors committed in Cantor's proof, specifically in regards to his use of the diagonal method and the Reductio Ad Absurdum proof.


Also, I fail to see how the one might mount an attack of God's omniscience from the premise of the uncountability of R. As it appears to me, if it is impossible to ennumerate R, then it is impossible to know every element of R. Where God's omniscience is defined as "knows all that is possibly knowable," and if the cardinality of R is unknowable in a real sense, then God's omniscience is still preserved--there just exist some things that are unknowable even for God.

There are a couple of things in regard to this. Here is the idea behind the proof against God's omniscience, and against God's existence...

1. If God exists, then God is omniscient.
2. If God is omniscient, then, by definition, God knows [the set of] all truths.
3. If Cantor’s theorem is true, then there is no set of all truths.
4. But Cantor’s theorem is true.
5. Therefore, God does not exist.

You object to premise 2 on the basis of your definition of "omniscience." Not everyone agrees to your definition. The idea that there exists some things that are unknowbale even for God goes against orthodoxy and even begins to attack other core doctrines such as God's omnipotence. For example, how can something exist and God not know it? Does that mean God is not the creator of all that exists? The point is, your argument against premise 2 is based on a questionable definition. I think a better way to attack the proof is to attack premise 3 on the basis that God's being and His knowledge transcends creation. Therefore, to say that God cannot "know" the set of all truths because creation puts a limit on what's knowable does not follow. Of course, this assumes that R is part of creation. However, with all of that said, I believe the weakest part of the argument is premise 4. Cantor's theorem is not true.


BTW - As at least one Tweb participant will tell you (IIRC his real-life work involves structural engineering) sand actually makes a rather reliable foundation...

I was playing off of Matthew 7:26 which reads, "Everyone who hears these words of Mine and does not act on them, will be like a foolish man who built his house on the sand." Verse 27 ends with, "...and great was its fall."

Again, thanks for your comments.

Sincerely,

Brian

[Brian]

Hello Rand!

Thank you for your post. I will be looking forward to your response when you get back home from work.


Of course, there are other ways to get there as well.

The only proof that R is uncountable is Cantor's proof utilizing his diagonal method. The only basis for the transfinite is this same proof.


First we would establish the theorem that says no set is equivalent to its power set (I trust you have no problem with this result).

This is a little circular in that the proof that |N*|>|N| utilizes RAA and diagonalization, and commits the same errors as does the uncountability theorum does. Therefore, I would argue that |*N| is not >|N| for the same reasons |R| is not >|N|. Again, Cantor's uncountability proof is the only basis for the uncountability of R and the transfinite.

Sincerely,

Brian

[Rand]

Today @ 08:29 PM post located here (http://www.theologyweb.com/forum/showthread.php?s=&postid=132472#post132472)
Brian:

This is a little circular in that the proof that |N*|>|N| utilizes RAA and diagonalization, and commits the same errors as does the uncountability theorum does. Therefore, I would argue that |*N| is not >|N| for the same reasons |R| is not >|N|. Again, Cantor's uncountability proof is the only basis for the uncountability of R and the transfinite.



Question 1: What is wrong with using reductio ad absurdum?

Question 2: Once again, what are these so-called problems with Cantor's diagonalization proofs?

By the way, I was not worried about proving that the natural numbers are not quivalent to the power set of the natural numbers. It can be proven that no set is equivalent to its power set (using reductio ad absurdum).

One last thing: do you have a degree in mathematics? Just wondering...

Rand

[Brian]

Hello Rand!


Question 1: What is wrong with using reductio ad absurdum?

There is nothing wrong with RAA. As a side note there has been much discussion concerning the form of RAA utilized in Cantor's proof. The traditional RAA proof arrives at a contradiction from outside the proof itself, whereas in Cantor's proof, Cantor derives the contradiction from within the proof. Here is what I mean...

Traditional RAA (RAA)

Prove: A
Assume: ~A
~A-->B (which we know contradicts a definition, theorem, etc...from outside the proof) That is to say:
~B is true from an outside the proof source.
Therefore, by Modus Tollens ~~A, which is A. QED.

Cantor's RAA (CRAA)

Prove: A
Assume: ~A
~A-->B
B-->~B
Therefore, by Modus Tollens ~~A, which is A. QED.(?)

The validity of this questionable. What exatly is the proving force of B-->~B? Is it any different from B-->B? This is refered to as the unproven base error, or circulus vitiosus.


Question 2: Once again, what are these so-called problems with Cantor's diagonalization proofs?

There are several problems with the proof, one of which is this circulus vitiosus. Other issues include the validity and applicability of the Standard Definition for countability utilized by Cantor, his unstated assumtion that all sets are actual, and his unstated assumtion that the only enumeration of R by N allowed is an enumeration that utilizes every element of N.


By the way, I was not worried about proving that the natural numbers are not quivalent to the power set of the natural numbers. It can be proven that no set is equivalent to its power set (using reductio ad absurdum).

OK. You said earlier...
First we would establish the theorem that says no set is equivalent to its power set (I trust you have no problem with this result). Then, we can show that the set of real numbers is equivalent to the power set of the natural numbers. Thus, the natural numbers are not equivalent to the real numbers (i.e. the set of real numbers is uncountable)

How do propose to prove that |R|=|N*| apart from Cantor's proof that |R|> than aleph_0? Also, I would like to see your proof that the cardinality of the power set of an arbitrary set has a greater cardinality than the original arbitrary set, or |A*| > |A|. I think it will suffer from the same problems that Cantor's proof suffers from.


One last thing: do you have a degree in mathematics? Just wondering...

No, I do not.

Sincerely,

Brian
P.S. Any thoughts on the debate between Traditional and Modern logicians concerning existential import?

[Rand]

Sorry Brian--

I had a good reply for you, but I was timed out when I hit "Submit," so I lost it. I will put it up in a little while, though.

Rand

[Dee Dee Warren]

Ahh the board monster ate it... I always type my responses in a word processor and paste into the reply box.

[Rand]

Yesterday @ 10:23 PM post located here (http://www.theologyweb.com/forum/showthread.php?s=&postid=132556#post132556)
Brian:
...and his unstated assumtion that the only enumeration of R by N allowed is an enumeration that utilizes every element of N.

Follow up on this statement. In set theory, set equivalence has a very specific meaning (as does countable). Before I go into your statement further, I would like to see what you mean.


Yesterday @ 10:23 PM post located here (http://www.theologyweb.com/forum/showthread.php?s=&postid=132556#post132556)
Brian:

How do propose to prove that |R|=|N*| apart from Cantor's proof that |R|> than aleph_0? Also, I would like to see your proof that the cardinality of the power set of an arbitrary set has a greater cardinality than the original arbitrary set, or |A*| > |A|. I think it will suffer from the same problems that Cantor's proof suffers from.

Theorem: No set is equivalent to its power set.
Proof: Let A be a set. Its power set will be denoted P(A). Let g be a function from A to P(A). Let B be the set of all x in A such that x is not in g(x). We claim that B is not in the range of g, and this can easily be shown; suppose that there is a y in A such that g(y)=B. Now, there are two cases:
If y is an element of B, then by the definition of B, y is not an element of g(y). Since g(y)=B, this means that y is not an element of B, which is a contradiction.
If y is not an element of B, then y is not an element of g(y), but since g(y)=B, this means y is not an element of B, which is a contradiction.
So, indeed, there is no y such that g(y)=B, and we have that B is not in the range of g. Therefore, since g is not an onto function, A and P(A) are not equivalent.

Now, this result tells us that there are uncountable sets-- the power set of the natural numbers being a good example. The other proof (that the set of real numbers is equivalent to the power set of natural numbers) can be shown by writing the real numbers in their binary form and constructing a binary string for each subset of the natural numbers. For example, the sets {1,2,5,7,8} and N\{3} can be viewed as
110010110000...
110111111111...,
respectively, and the one-to-one correspondence is obvious.

I am not sure what the existence of uncountable sets has to do with arguments for or against the existence of God... I saw the "argument" you posted in response to garthoverman, but that is so full of holes that it is hardly worth the time is takes to type it out.

Rand

By the way, I asked if you had a mathematics degree because there is a 'soft' proof for the uncountability of the real numbers in basic measure theory. It happens as follows:
All countable sets have measure 0 (an easily proved statement).
The real numbers has measure infinity (from the definition of LeBesgue Measure).
Therefore, the real numbers are not countable.

[Brian]

Hello Rand!

Thank you for such a thoughtful post. Before I begin, can we for the sake of simplicity assume that when we are talking about sets, unless mentioned otherwise, we are speaking of infinite sets? In terms of finite sets, I acknowledge that many theorems are trivially valid, and the issues seem to concern infinite sets only. Also, I have some questions concerning your post that probably arise from my lack of mathematical training. I have not heard of basic measure theory, and consequently did not follow your "soft proof" for the uncountabilty of R. However, I would prefer to table that discussion for later if you don't mind. As to your proof that no set is equivalent to it's power set, its RAA form would seem to follow this:

Prove A: |A|=/=|P(A)|
Assume~A: |A|=|P(A)|
~A-->B: There exists a "bijection" between A and P(A).

You have said it this way...


Let g be a function from A to P(A).

Is this the same as saying: There exists a function g, such that if x is an element of A then g(x) is an element of P(A)? Also, I would ask how do you know that if two sets have the same cardinality, then there exists a bijection between them? As far as I am aware this has never been proved.

You next say...


Let B be the set of all x in A such that x is not in g(x).

Is this the same as saying: Let B be the set of all x in A such that g(x) is not an element of P(A)? And if so, according to our assumtion ~A, then wouldn't that mean B contains no elements? Like I said, this all may be a misunderstanding on my part due to my lack of education. I will wait for your clarification before I continue.

Thanks!

Brian

[Rand]

Today @ 07:05 PM post located here (http://www.theologyweb.com/forum/showthread.php?s=&postid=133304#post133304)
Brian:

Is this the same as saying: There exists a function g, such that if x is an element of A then g(x) is an element of P(A)? Also, I would ask how do you know that if two sets have the same cardinality, then there exists a bijection between them? As far as I am aware this has never been proved.


As to the first part, I think you are right. g is simply a function from A to P(A), so for every x in A, g(x) is an element of P(A) (in other words, g(x) is a subset of A).

For the second part, the definitions of set equivalence and cardinal number tell us that a bijection exists between two sets with the same carindality. In an axiomatic development of set theory, this gets pretty ugly, in that cardinal numbers are defined in terms of ordinal numbers, but rather than go into that, we can give a more naive approach to sets.

We can think of the cardinal number of a set as an arbitrary representative of a class of mutally equivalent sets. We define two sets to be equivalent if there is a bijection between them. Note than the concept of number, as we are used to it, is not necessary for set comparisons.



Is this the same as saying: Let B be the set of all x in A such that g(x) is not an element of P(A)? And if so, according to our assumtion ~A, then wouldn't that mean B contains no elements? Like I said, this all may be a misunderstanding on my part due to my lack of education. I will wait for your clarification before I continue.

No-- B is not the set of all x in A such that g(x] is not an element of P(A), because like we said, the range of g is in P(A). B is the set of all x in A such that x is not an element of g(x). Perhaps it will be more clear this way: B is the set of all x in A such that x is not a member of the set that g maps it to.

Look at this example:
A = {1,2}
P(A) = {0,{1},{2},{1,2}} (0 is the null set)

Now define g as follows:
g(1) = {1,2}
g(2) = {1}

In this instance, B={2}, because 2 was mapped to the set {1}, but 2 is not an element of {1}.

Soemitmes seeing what B actually is for the first time can be tricky, especially with no background in the material, but once it clicks, the proof is easy.

Rand

[Brian]

Hello Rand!

Thank you for your clarification on this: B is the set of all x in A such that x is not an element of g(x). Can you explain to me what you mean by "we claim that B is not in the range of g"? Is this 'range' the set of all g's for every x? In other words, in your previous example is the range of g={{1,2},{1}}? I appreciate your patience here.

I would like to make some observations concering your proof as it relates to N. For our purposes I'll define N as {1,2,3,...n,...}. I would appreciate your comments on this:

1. If |P(N)|=/=|N|, then |P(N)|>|N|, and P(N) is uncountable.
2. Since we are speaking in terms of mappings, then it must be true that there exists an element of P(N) that is not mapped to by an element of N. This means, at some point in our listing all of the elements of N have been exhausted, and there are still elements left over in P(N).
3. Because of Cantor's Countable Union Theorem (i.e. the union of two countable sets is countable), then it follows that the set of elements not mapped to must themselves be uncountable.
4. Points 1,2,3 must necessarily follow as a consequence of your theorem.


We define two sets to be equivalent if there is a bijection between them. Note than the concept of number, as we are used to it, is not necessary for set comparisons.

I am going to discuss this in terms of countability, which is what started this thread. Here is the commonly accepted definition for countable...

The Standard Definition of Countable (SD): A set is countable iff it can be placed into bijection with N.

This can be broken down in to the following 2 implications:

(1) If a set can be placed into bijection with N, then it is countable.
(2) If a set it countable, then it can be placed into bijection with N.

It is intersting to note that these two implications are logically independent. That is, if (1) is true, it does not automatically mean (2) is true, or vice versa. Therefore, in order to satisfy SD, both (1) and (2) must be shown to be true.

Statement (1) I find to be obviously true. For the sake of distinction, I would like to call (1) the "Definition of Countable (DOC)" even though it is only an implication.

Statement (2) is logical, but it is logically independent of (1). In other words, the validity of DOC does not imply the validity of (2). The contrapositive of A-->B is ~B-->~A, and is logically equivalent to A-->B. The contrapositive of (2) is "if a set cannot be placed into bijection with N, then it is uncountable." Again, for the sake of distinction, I would like to refer to this as the "Definition of Uncountable (DOU)." That would make (2) the contrapositive of DOU, i.e. CDOU. Therefore, SD consists of 2 logically independent implications, DOC and DOU. From this we now have a definition that provides a sufficient condition to determine a set to be countable or uncountable. That is, DOC establishes a sufficient condition to determine that a set is countable, and DOU establishes a sufficient condition to determine that a set is uncountable.

DOC: If a set can be placed into bijection with N, then it is countable.
DOU: If a set cannot be placed into bijection with N, then it is uncountable.

Any comments on this?

Sincerely,

Brian

[Rand]

Today @ 01:55 AM post located here (http://www.theologyweb.com/forum/showthread.php?s=&postid=133614#post133614)
Brian:

Thank you for your clarification on this: B is the set of all x in A such that x is not an element of g(x). Can you explain to me what you mean by "we claim that B is not in the range of g"? Is this 'range' the set of all g's for every x? In other words, in your previous example is the range of g={{1,2},{1}}? I appreciate your patience here.

I think you have the concept of range correct. In my example, the range was {{1,2},{1}}, although the general explanation you gave is not correct. The range of the function g is not the set of all g's for every x. It is the set of all g(x), where x is an element of A. I think you meant to say this, but I wanted to make sure.


1. If |P(N)|=/=|N|, then |P(N)|>|N|, and P(N) is uncountable.
2. Since we are speaking in terms of mappings, then it must be true that there exists an element of P(N) that is not mapped to by an element of N. This means, at some point in our listing all of the elements of N have been exhausted, and there are still elements left over in P(N).
3. Because of Cantor's Countable Union Theorem (i.e. the union of two countable sets is countable), then it follows that the set of elements not mapped to must themselves be uncountable.
4. Points 1,2,3 must necessarily follow as a consequence of your theorem.

I think you are correct here as well (although we can go a bit further and say that the countable union of countable sets is countable). :)




I am going to discuss this in terms of countability, which is what started this thread. Here is the commonly accepted definition for countable...

The Standard Definition of Countable (SD): A set is countable iff it can be placed into bijection with N.


A better definition of countable (and one that you would find in a set theory book) would be the following: a set is countable iff its cardinal number is less than or equal to Aleph_0.

In the case of infinite sets, this would reduce to your definition-- an infinite set is countable iff there is a bijection from it to N.

The rest of your post did not make much sense to me-- I followed what you were saying (sort of), but did not see the point.

Having already proven that the main question of this thread, I fail to see what the next step may be. Let me know...

[Brian]

Hello Rand!


The rest of your post did not make much sense to me-- I followed what you were saying (sort of), but did not see the point.

Forgive me for being so cryptic. I have not gotten to the point as of yet, and it will take me at least another post until I do. However, what comes next should help shed some light on its relevance to the topic at hand. Here is the beginning of Cantor's proof:

Prove A: R is uncountable.
Assume ~A: R is countable.
~A-->B: There exists a bijection between R and N.

Now let's look at the beginning of your proof:

Prove A: P(A)=/=A
Assume ~A: P(A)=A
~A-->B: There exists a bijection between P(A) and A. (Note: you said it this way...let g be a function from A to P(A).)

I would like to reword your proof in terms of infinite sets. I do not have an issue with the proof in terms of finite sets. Also, I would like to do this relative to N because the cardinality of N defined as aleph_0 is the basis for our definition of countability. This you yourself have pointed out. This leads us the following equivalent representation:

Prove A: P(N) is uncountable.
Assume ~A: P(N) is countable.
~A-->B: There exists a bijection between P(A) and A.

Now we have already defined DOU:If a set cannot be placed into bijection with N, then it is uncountable. I want you to notice that both you and Cantor are using the contrapositive of DOU as the basis for the implication ~A-->B, i.e., if a set is countable, then it can be placed into bijection with N. I also want you to notice that in no way does either proof rely on DOC. Are you with me so far?

I missed this and wanted to comment on it. You said earlier...


I saw the "argument" you posted in response to garthoverman, but that is so full of holes that it is hardly worth the time is takes to type it out.

I assume the "argument" you are referring to is the one put forth by those who wish to disprove God's existence. I have already put forth where I thought there were difficulties. If you don't mind, I would very much like to get your take on the difficulties.

Thank you so much for your consideration.

Sincerely,

Brian

[Rand]

I will comment on the "argument" against God's existence like you asked in the previous post rather than continue with the mathematics discussion until you are able to actually show a flaw in the argument. I await your further illumination on the subject, although I have a feeling the discussion may lead nowhere.


06-25-2003 @ 08:09 PM post located here (http://www.theologyweb.com/forum/showthread.php?s=&postid=132464#post132464)
Brian:

1. If God exists, then God is omniscient.
2. If God is omniscient, then, by definition, God knows [the set of] all truths.
3. If Cantor’s theorem is true, then there is no set of all truths.
4. But Cantor’s theorem is true.
5. Therefore, God does not exist.

I have no (real) objections to (1) or (2), although (2) could probably be worded differently. Statements (3) and (4) are where the trouble arrives.

The trouble with the argument is that its author has obviously never studied advanced mathematics. In an axiomatic development of set theory, we only deal with sets. There are no numbers (other than the numbers we create in the theory itself-- using sets); there are no shapes, figures, characters, and more importantly, there are no "truths" (left undefined in the argument, of course). The only "truths" in set theory are the theorems that we can prove using the basic axioms (of either ZFC set theory, the set theory of Von Neumann, Godel, and Bernay, or even the system of Quine). All of our sets in set theory only contain other sets-- they do *not* contain "truths." So in reality, the author is using a clever form of equivocation when he talks about "the set of all truths." He wants "set" to be the same things we talk about in set theory, but he wants to fill them up with something that does not belong to the same set theory.

Of course, this argument fails even if we allow the distortion of "set" to remain. In some axiomatic developments of set theory, there are ways to talk about all sets-- the use of classes being the most common way. So if the author really wants to use mathematics (in a very unmathematical way), why not stick all of these undefined "truths" in a class?

The author's main error is that he expects God to use mathematics, which is only an abstraction of the human mind. If he wants to get technical, no sets exist. They are all created by man. So, when the author can no longer abuse set theory, the word "set" in his argument becomes something more basic, and then we have no trouble imagining "the set of all truths."

I think this argument must have been made by someone who heard of Russell's paradox, but never took the time to understand it.

Rand

[Brian]

Hello Rand!


All of our sets in set theory only contain other sets-- they do *not* contain "truths." So in reality, the author is using a clever form of equivocation when he talks about "the set of all truths." He wants "set" to be the same things we talk about in set theory, but he wants to fill them up with something that does not belong to the same set theory.

I am not sure I understand. Let's take for instance the set of natural numbers. What is the nature of the members of this set?


The author's main error is that he expects God to use mathematics, which is only an abstraction of the human mind.

How do you know this? Why not consider mathematics an abstraction of the way God thinks?


If he wants to get technical, no sets exist. They are all created by man.

Once again, how do you know this? Could we define a set such that each element is a different commandment from the 10 commandments found in Exodus?


So, when the author can no longer abuse set theory, the word "set" in his argument becomes something more basic, and then we have no trouble imagining "the set of all truths."

According to you, what is a "set"? Rand, I am not trying to defend the author's position. I am just trying to understand the points you are making. Forgive me for my naivete.

OK, on to one of the issues that I have with Cantor's and your proof. I have made the case that Cantor's and your proof rely on DOU. DOU claims that "the set of all sets that have the same cardinality as N," and "the set of all sets that are able to be put into bijection with N" is the same set. However, this has never been proved.

DOC and DOU are logically independent of one another. It is obvious to me that DOC is true, but even if it is true it does not necessarily follow that DOU is true. Because the DOC and the DOU are logically independent it must be shown that these 2 definitions are consistent with each other. There is no proof that I am aware of that demonstrates the absolute consistency of the DOC and DOU with respect to all mapping techniques and all possible infinite sets no matter what the differences in properties between the 2 sets.

In other words, Cantor's and your proof are operating under a second assumption, namely DOU. As I am sure you are aware, an RAA proof with more than one assumption does not prove anything. Therefore, Cantor's proof fails to prove R is uncountable, and your proof fails to prove that P(N) is uncountable.

I await your comments.

Sincerely,

Brian

[Rand]

Yesterday @ 10:46 PM post located here (http://www.theologyweb.com/forum/showthread.php?s=&postid=134582#post134582)
Brian:

I am not sure I understand. Let's take for instance the set of natural numbers. What is the nature of the members of this set?

The elements of the set of natural numbers are sets. There are several ways to define them, but here is the most common way:

0 = {} (empty set)
1 = {{}} (the set containing the empty set)
2 = {{},{{}}} (the set containing 0 and 1)
3 = {{}.{{}},{{},{{}}}} (the set containing 0,1,2)
.
.
.
So a natural number, x is the set containing all of the natural numbers less than x (and we define 0 to be the null set).

Defining the natural numbers in this way has the funny property that every natual number is a subset of the next natural number and every natural number is also a member of the next natural number.


How do you know this? Why not consider mathematics an abstraction of the way God thinks?

Once again, how do you know this? Could we define a set such that each element is a different commandment from the 10 commandments found in Exodus?

According to you, what is a "set"? Rand, I am not trying to defend the author's position. I am just trying to understand the points you are making. Forgive me for my naivete.

In regards to your first question; I cannot consider mathematics to be an abstraction of the way God thinks because it is invented by man. Yes, it fits together nicely and can be quite beautiful at times, but I have trouble believing that God just happens to work the same way as this man-created system. Besides, mathematics cannot give us all truths (think Godel), so if God is mathematical in nature, then He must be as confused as we are.

To answer your second and third questions: if you can develop an axiomatic system of set theory where sets are defined to be the different commandments in the Decalogue, then more power to you. I, however, and the rest of the mathematicians on the planet will not give a definition to "set." In an axiomatic development of set theory, "is a set" is primitive (along with "is an element of"). We may abstractly think of a set as "a collection of objects" or something of that nature, but when we get down to the nuts and bolts of the system, we end up leaving it undefined.


OK, on to one of the issues that I have with Cantor's and your proof. I have made the case that Cantor's and your proof rely on DOU. DOU claims that "the set of all sets that have the same cardinality as N," and "the set of all sets that are able to be put into bijection with N" is the same set. However, this has never been proved.

Well, you are wrong in this matter. My proof does the following:

If we look at the sets N and P(N), it can be proven that any function from N to P(N) is not onto. I proved this. You can extract this small fact from the rest of the proof I originally gave and note that it does not say anything about countable or uncountable sets.

Now, since any function between this two sets is not onto, there can be no bijection between them (since a bijection is a one-to-one, onto function). Thus, these sets are not equivalent (note I have still not used anything about countable or uncountable sets). So, if we look at the class of sets that are equivalent to N and the class of sets that are equivalent to P(N]), we are looking at two different classes.

At this point, we have shown that the cardinal numbers of N and P(N) are different (from the naive definition of cardinal number I gave a few posts back). Lets say N has cardinal number a (in reality Aleph_0) and P(N) has cardinal number b. Now, any two cardinal numbers can be compared, so one of the following is true:
a = b a<b a>b

It is obvious that a>b is false, and we already know that a=b is false, so it must be true that a<b. NOTE: WE STILL HAVE NOT USED THE DEFINITIONS OF COUNTABLE OR UNCOUNTABLE SETS.

Now, I will define a countable set:

Definition: A set is countable iff its cardinal number is less that or equal to a.(1)

Thus, P(N) is not countable. If you would like to argue with this result, you do not even need to know what a countable set is until the last step, so you will have to show how a function from N to P(N) can be onto (I proved this could not happen).

Have fun.

(1) This definition is perfect valid. See Enderton, Elements of Set Theory, 1977 Academic Press, page 159.

[HRG_new]

Yesterday @ 09:24 PM post located here (http://www.theologyweb.com/forum/showthread.php?s=&postid=134510#post134510)
Rand:

I will comment on the &quot;argument&quot; against God's existence like you asked in the previous post rather than continue with the mathematics discussion until you are able to actually show a flaw in the argument. I await your further illumination on the subject, although I have a feeling the discussion may lead nowhere.



I have no (real) objections to (1) or (2), although (2) could probably be worded differently. Statements (3) and (4) are where the trouble arrives.

The trouble with the argument is that its author has obviously never studied advanced mathematics. In an axiomatic development of set theory, we only deal with sets. There are no numbers (other than the numbers we create in the theory itself-- using sets); there are no shapes, figures, characters, and more importantly, there are no &quot;truths&quot; (left undefined in the argument, of course). The only &quot;truths&quot; in set theory are the theorems that we can prove using the basic axioms (of either ZFC set theory, the set theory of Von Neumann, Godel, and Bernay, or even the system of Quine). All of our sets in set theory only contain other sets-- they do *not* contain &quot;truths.&quot; So in reality, the author is using a clever form of equivocation when he talks about &quot;the set of all truths.&quot; He wants &quot;set&quot; to be the same things we talk about in set theory, but he wants to fill them up with something that does not belong to the same set theory.

Of course, this argument fails even if we allow the distortion of &quot;set&quot; to remain. In some axiomatic developments of set theory, there are ways to talk about all sets-- the use of classes being the most common way. So if the author really wants to use mathematics (in a very unmathematical way), why not stick all of these undefined &quot;truths&quot; in a class?

The author's main error is that he expects God to use mathematics, which is only an abstraction of the human mind. If he wants to get technical, no sets exist. They are all created by man. So, when the author can no longer abuse set theory, the word &quot;set&quot; in his argument becomes something more basic, and then we have no trouble imagining &quot;the set of all truths.&quot;

I think this argument must have been made by someone who heard of Russell's paradox, but never took the time to understand it.

Rand

Hi Rand!

However, the Power Set theorem indicates that the concept of omnipotence may be self-contradictory.

1. Let A be the set of all actions that God can take.

2. Let T(S), S a subset of A, be the action "thinking simultaneously about all actions in S, but about no others". This is a logically consistent action; thus if God is omnipotent, it can be identified with an element of A.

3. T is an injection (different S result in different T(S) ) from A into P(A).

4. Contradiction because of Power Set theorem.

Regards,
HRG.

[Rand]

Hello HRG--

I thought that you might make an appearance on this thread. From your posts on CARM, its seems like a subject you know quite a bit about (more than me, anyway). :)

Are you still posting on CARM, or did you abandon ship in lieu of the moderators actions over the past couple of weeks?


Today @ 12:01 PM post located here (http://www.theologyweb.com/forum/showthread.php?s=&postid=134881#post134881)
HRG_new:
However, the Power Set theorem indicates that the concept of omnipotence may be self-contradictory.


...that is why I have trouble assigning such human concepts to God without the provision that they are just attempts to describe something that cannot be described as it actually is.

With such a strict view of omnipotence, I would have to say that God is not omnipotent. However, once we understand that God's 'omnipotence' only scratches the surface of what He is, then there are no problems, and the arguments put forth in this thread no longer matter.

...though I did like your argument as an exercise with sets. :)

The thing that really interested me about this thread was a problem that I have seen on the EVO board on CARM. It seems that when some people see an idea that possibly distorts their view of God, they reject it immediately. We see this often with people who try to find some way to accept a young-earth model. Perhaps that can be understood, since evolution gets so much negative press in evangelical circles and there is so much misinformation out there. But when basic mathematical truths are thrown out on a whim, it kind of scares me. What think you?

Glad to see you here, in any event.

[Brian]

Hello Rand!


In regards to your first question; I cannot consider mathematics to be an abstraction of the way God thinks because it is invented by man.

I am not sure what you mean by this. What do you mean by man’s invention? Could mathematics be the discovery of something that is already built into creation? Man is made in the image of God. Could mathematics be part of the reflection of that image? Also, if man invented mathematics, does that mean God is not the creator of all things?


Yes, it fits together nicely and can be quite beautiful at times, but I have trouble believing that God just happens to work the same way as this man-created system.

Again, if we were made in God’s image, then would it be possible for the way we think to reflect the way God thinks at some level? If God were the creator, wouldn’t this be a possibility?


Besides, mathematics cannot give us all truths (think Godel), so if God is mathematical in nature, then He must be as confused as we are.

Your argument against mathematics being an abstraction of the way God thinks seems to be:

Mathematics cannot give us all truths.
God’s is mathematical in nature.
Therefore, God cannot know all truths.

In order to support your presupposition it seems as if you are using a similar argument to the one we are refuting. Also, God’s being mathematical in nature does not necessarily mean that God is limited by our understanding of mathematics.


Well, you are wrong in this matter. My proof does the following:

If we look at the sets N and P(N), it can be proven that any function from N to P(N) is not onto. I proved this.

Rand, I am obviously not following. Let me try and understand. You are proving to me that |P(N)|=/=|N|. To do this, you have presented a proof utilizing the reductio ad absurdum (RAA) method. The first step in the RAA proof is:

Prove A: |P(N)|=/=|N|
The second step is to assume the negation of what you want to prove. That is,
Assume ~A: |P(N)=|N|
The very next step in your proof was…


Let g be a function from A to P(A).

I then asked you to clarify this by saying…


Is this the same as saying: There exists a function g, such that if x is an element of A then g(x) is an element of P(A)?

To which you said…


As to the first part, I think you are right. g is simply a function from A to P(A), so for every x in A, g(x) is an element of P(A).

So, you are saying:

~A-->B: There exists a function g that maps every element in N to P(N).

You then define B as the set of all x in A such that x is not in g(x). And then you say this...


We claim that B is not in the range of g...

To establish this claim you derive a contradiction. What is it exactly you are contradicting? Are you contradicting the assumtion that B is in the range of g? If so, what gives you the right to assume that B should be in the range of g? Could the answer be that you are assuming a bijection exists between P(N) and N based on the assumption |P(N)|=|N|?

SD follows from the definition you proposed for “countable.” Why do you think SD is valid? What is your proof? I have raised an objection to it, and you have just turned around and asserted that it is valid. As I have already mentioned, SD is made up of 2 implications that are logically independent. That means they must be consistent with each other in order for SD to be valid. There has never been a proof demonstrating the consistency between DOU and DOC. Perhaps you can help me out here.

Sincerely,

Brian

Edited to add:

Here might be a counter example to your theorem. Let’s consider the set of prime numbers. Now we know that this set is a subset of N, i.e. all of the elements in P are in N. I assume you will also have no problem agreeing that the cardinality of P is equal to N. Based on these assumptions let’s attempt an ordered listing of P(P): {0, {2}, {3}, {2,3}, {5}, {2,5}, {3,5}, {2,3,5}, {7}, {2,7}, {3,7}, {2,3,7}, {5,7}, {2,5,7}, {3,5,7}, {2,3,5,7}, etc…}.The method being used to list P(P) should be obvious, and equally obvious is that every element of P(P) will be in the list once the work is done. An interesting thing occurs at this point. I can put this ordered listing into a one-to-one correspondence with N! I start with 0 being paired with 1. Then {2} is paired with 2. {3} is paired with 3. {2,3} is paired with 6, i.e. (2x3). {5} is paired with 5. {2,5} is paired with 10, i.e. (2x5)….so on and so forth. Because of the unique property of prime numbers, when this work is done no element of N will be used twice, and every element from P(P) will have been enumerated by N. This means |P(P)|=|N|! I am sure you see the difficulty.

[Rand]

Yesterday @ 07:30 PM post located here (http://www.theologyweb.com/forum/showthread.php?s=&postid=135036#post135036)
Brian:

Rand, I am obviously not following. Let me try and understand. You are proving to me that |P(N)|=/=|N|. To do this, you have presented a proof utilizing the reductio ad absurdum (RAA) method. The first step in the RAA proof is:

No-- I only want to prove one thing. I want to show that if you have a set A and its power set, P(A), then any function g that maps A to P(A) is not onto. This has nothing to do with cardinal numbers, countable sets, or uncountable sets. We only need the definition of "function" and "onto."

From the questions you psed (that I did not quote), it seems like you did not follow the proof very well. So, more clearly:

Theorem: Let A be a set. We will denote its power set by P(A). Let g be a function from A to P(A). Let B be the set of all x in A such that x is not an element of g(x). The existence of this set is guaranteed by the axiom schema of separation (since B is a subset of A, it is an element of P(A)). We will show that g is not onto.

Proof:For the sake of contradiction, assume that g is onto. Since g is onto, there exists a y in A such that g(y)=B (definition of onto). Now, if y is an element of B, then by the rule that defines B, y is not an element of g(y). But g(y)=B, so y is not an element of B, which is a contradiction. Thus, y is not an element of B. Now, if y is not an element of B, then y is not an element of g(y). But by the definition of B, this means that y is in B, which is a contradiciton. Therefore, there is no y such that g(y)=B, and we have that g is not onto. You seem to like formal arguments, so here it would be:

Q: g is onto
R: there exists a y such that g(y)=B
Assume Q
Q->R (definition of onto)
R->~R (definition of the set B)
(Q->R).(R->~R)->~Q (draw a truth table-- it works)
Therefore, g is not onto.


SD follows from the definition you proposed for “countable.” Why do you think SD is valid? What is your proof? I have raised an objection to it, and you have just turned around and asserted that it is valid.

It is not necessary that I prove a definition-- that is why it is called a definition. That is the way mathematicians define "countable." From a very naive view, we only need the definition of "cardinal number" to understand what countable means.

Never mind anything about countable or uncountable sets at this point. If you have a problem with the above proof, let me know. Once this is established, then we can move on.


Here might be a counter example to your theorem. Let’s consider the set of prime numbers. Now we know that this set is a subset of N, i.e. all of the elements in P are in N. I assume you will also have no problem agreeing that the cardinality of P is equal to N. Based on these assumptions let’s attempt an ordered listing of P(P): {0, {2}, {3}, {2,3}, {5}, {2,5}, {3,5}, {2,3,5}, {7}, {2,7}, {3,7}, {2,3,7}, {5,7}, {2,5,7}, {3,5,7}, {2,3,5,7}, etc…}.The method being used to list P(P) should be obvious, and equally obvious is that every element of P(P) will be in the list once the work is done. An interesting thing occurs at this point. I can put this ordered listing into a one-to-one correspondence with N! I start with 0 being paired with 1. Then {2} is paired with 2. {3} is paired with 3. {2,3} is paired with 6, i.e. (2x3). {5} is paired with 5. {2,5} is paired with 10, i.e. (2x5)….so on and so forth. Because of the unique property of prime numbers, when this work is done no element of N will be used twice, and every element from P(P) will have been enumerated by N. This means |P(P)|=|N|! I am sure you see the difficulty.

Sorry-- you have not presented a counterexample. Look in your list again; where is the set of all prime numbers? In your list, every set you present is finite. Once you place the set of all primes in the list (and all infinite subsets of that), your list will be uncountable.

You will not be able to get a counterexample to my* theorem, but keep trying, if you wish. :)

*calling it "my" theorem is not correct. These basic facts have been known for over a 100 years, after all.

[Brian]

Hello Rand!

This post is only going to deal with some side issues that I felt needed to be addressed. Concerning the argument against the omniscience (and existence) of God, you have said…


I saw the "argument" you posted in response to garthoverman, but that is so full of holes that it is hardly worth the time is takes to type it out.

After which I asked you to go ahead and give me your critique. You started with…


In an axiomatic development of set theory, we only deal with sets. There are no numbers (other than the numbers we create in the theory itself-- using sets); there are no shapes, figures, characters, and more importantly, there are no "truths" (left undefined in the argument, of course).

The idea behind your argument seemed to be that the author had commited a “subtle equivocation” concerning his use of “set”. However, HGR_New stepped in and seemingly contradicted this argument by putting forth one of his own based on the Power Set Theorem. The first statement in his argument puts all of God’s possible actions into a set A. I fully expected you to take issue with this, but instead you denied the Christian orthodox view of omniscience, and bowed to his expertise…


From your posts on CARM, its seems like a subject you know quite a bit about (more than me, anyway)...that is why (HGR's Power Set Argument against the orthodox understanding of omnipotence) I have trouble assigning such human concepts to God without the provision that they are just attempts to describe something that cannot be described as it actually is…with such a strict view of omnipotence, I would have to say that God is not omnipotent.

It seems to me that your argument of equivocation is not very strong. Next, you made several assertions, which were more philosophical in nature. Once again, I have asked you the basis for your assertions. In your last post you choose not to answer my “how do you know” questions. Based on what you have said thus far, I do not think you have demonstrated that the “argument is full of holes.”

Then your response to HGR_New ended with…


The thing that really interested me about this thread was a problem that I have seen on the EVO board on CARM. It seems that when some people see an idea that possibly distorts their view of God, they reject it immediately…. But when basic mathematical truths are thrown out on a whim, it kind of scares me.

There are a couple of issues here. One is you seem to have assumed that the reason I take issue with Cantor’s proof is because Cantor’s proof creates problems with my view of God. That is only partially true. I agree that if R is uncountable, then the argument we have been discussing becomes much stronger. It was evident in your response to HGR_New that you felt the force of this as well. However, before I even knew of the epistemological ramifications of the uncountability theorem, I was convinced that Cantor’s proof contained logical errors and was not valid.

This brings up the second point, there is a storied list of mathematicians who reject the whole idea of the Transfinite, which is based on the idea of an actual infinity verses a potential infinity. Many great formulators of classical logic and mathematics such as Aristotle, Kant, Kronecker, Brouwer, Poincare, Leibniz, Gauss, Cauchy, etc., rejected the idea of actual infinity and readily embraced the concept of potential infinity. You may have heard of Aristotle’s famous statement, “Infinitum Actu Non Datur”, which roughly means that actual infinity is impossible. You speak of Cantor’s theorem as if it were a done deal. Even Wittgenstein (the pupil and colleague of Russell and Frege) was quoted as saying, “…Cantor’s argument has no deductive content at all.” So your characterization that Cantor’s theorem is a “basic mathematical truth” is a misrepresentation at best.

I will respond to your latest post probably tomorrow (Monday).

Sincerely,

Brian

[Rand]

Today @ 09:14 PM post located here (http://www.theologyweb.com/forum/showthread.php?s=&postid=135492#post135492)
Brian:
The idea behind your argument seemed to be that the author had commited a “subtle equivocation” concerning his use of “set”. However, HGR_New stepped in and seemingly contradicted this argument by putting forth one of his own based on the Power Set Theorem. The first statement in his argument puts all of God’s possible actions into a set A. I fully expected you to take issue with this, but instead you denied the Christian orthodox view of omniscience, and bowed to his expertise…

Perhaps you would get that impression from the exchange if you view it in a box. However, HRG and myself have both been active posters on another message board for quite some time, so I have some idea of his worldview and the purpose of his message.

I, however, did not deny the orthodox Christian position in my posts. "God's omnipotence connotes that He has the power to execute all that He may wish, that is, all that is real and possible." Furthermore, "As God's power is identical with God's Essence, it cannot imply anything which contradicts the Essence and the Attributes of God" (1).


It seems to me that your argument of equivocation is not very strong.

Of course it is-- when the author says "set," he derives a contradiction by viewing sets as a mathematician would. However, the only "set" he is dealing with is one that has no place in mathematics.


Next, you made several assertions, which were more philosophical in nature. Once again, I have asked you the basis for your assertions. In your last post you choose not to answer my “how do you know” questions.

That was a mistake on my part. I should not have allowed myself to go off on a tangent. I do not consider the questions necessary for a discussion of whether or not the real numbers are countable, so I will not address them any further.


There are a couple of issues here. One is you seem to have assumed that the reason I take issue with Cantor’s proof is because Cantor’s proof creates problems with my view of God. That is only partially true. I agree that if R is uncountable, then the argument we have been discussing becomes much stronger. It was evident in your response to HGR_New that you felt the force of this as well. However, before I even knew of the epistemological ramifications of the uncountability theorem, I was convinced that Cantor’s proof contained logical errors and was not valid.

I only responded to HRG's post as an aside. It really has no bearing on this conversation. And I do not lend any force to the "argument" posted in this thread against God's existence. It is a ludicrous attempt to hijack mathematics for an end that is not mathematical in nature.


So your characterization that Cantor’s theorem is a “basic mathematical truth” is a misrepresentation at best.

And how familiar are you with modern mathematics? You listed quite a few ancient and early-modern mathematicians along with some philosophers, who were born well before mathematics reached its current level of sophistication. Wittgenstein can perhaps be foriven his error, since set theory was not set on a firm axiomatic basis until well after Cantor.

Also, I do not really care about the argument with potential or actual infinities. In set theory, the existence of an infinite set is true from the axioms... If you haven't guessed by know, I am not terribly interested in philosophical hand-waving and hazy prose. One of the beautiful things about mathematics is that those things are not necessary. We have a list of axioms and rules for deduction, and it just so happens that the axioms tell us that uncountable sets do exist.

(1) Dr. Ludwig Ott, Fundamentals of Catholic Dogma, TAN Books 1960, page 47.

[Rand]

Just in case a lurker thinks Brian's position has any validity whatsoever, here is one more proof that the real numbers are uncountable. The methods used can be found in any mathematics course where measure theory is introduced.

Definition: Let A be a set of real numbers. Consider the sum of the lengths of of a countable number of intervals that cover A. The outer measure of A is defined to be the infimum of all such sums. The outer measure is denoted m(A).

From this definition it is clear that the outer measure of any interval is simply the interval length. The concept of outer measure can be refined, and it eventually gives us a really nice way to integrate functions (the Lebesgue Intergral) that cannot be done using the Reimann Integral. However, that is not important at this time, because just using the definition of outer measure, we can show that the real numbers are uncountable.

Theorem 1: For any countable collection of sets of real numbers, the outer measure of the union of those sets is less than or equal to the sum of the outer measure of each set. I will just state this wihout proof, but it is very easy to show. If anyone really wants a proof of it, just ask...

Theorem 2: Any countable set of real numbers has (outer) measure equal to 0.
Proof: Let A be a countable set of real numbers, {a1,a2,a3,...}. Let a be an element of A. For e>0, the interval (a-e,a+e) covers {a}. Since e was an arbitrary real number, {a} has outer measure 0. Similarly, any singleton point has outer measure 0. From theorem 1, the outer measure of A is less than or equal to the sum of the outer measure of each singleton element of A. Therefore,
[i]m(A)<= m({a1})+m({a2})+... = 0+0+...+0+... = 0
->m(A)=0
Corollary: The real numbers are uncountable.
Proof: Assume the real numbers are countable. By theorem 2, this means the real numbers have outer measure 0, but this is a contradiction, since no finite interval (nor a finite number of finite intervals) covers the reals (i.e. the outer measure of the real numbers is infinite). Thus, the real numbers are not countable.

[Brian]

Hello Rand!


Perhaps you would get that impression from the exchange if you view it in a box. However, HRG and myself have both been active posters on another message board for quite some time, so I have some idea of his worldview and the purpose of his message.

I was pointing out that his message seemed to contradict your argument of equivocation, and how mathematicians use “sets.”


I, however, did not deny the orthodox Christian position in my posts.

Rand, I certainly could have misconstrued what you were saying. You said…


With such a strict view of omnipotence, I would have to say that God is not omnipotent.

Would you mind explaining to me what this “strict view” entails?


Of course it is-- when the author says "set," he derives a contradiction by viewing sets as a mathematician would. However, the only "set" he is dealing with is one that has no place in mathematics.

How was this author’s use of sets different from HRG’s use?


I do not consider the questions necessary for a discussion of whether or not the real numbers are countable, so I will not address them any further.

That is fine. However, it was you that said that the author’s proof was full of “holes.” When you explained what you thought were “holes,” I raised several questions concerning your assertions. The issues are philosophical in nature rather than mathematical, and I can understand why you would rather not go there.


I only responded to HRG's post as an aside. It really has no bearing on this conversation.

At first I was not going to bring it up at all. However, in your post to HGR you inferred some things about the nature of this thread and my motivations. I felt it was appropriate to respond.


And I do not lend any force to the "argument" posted in this thread against God's existence. It is a ludicrous attempt to hijack mathematics for an end that is not mathematical in nature.

You have already asserted this. However, from a philosophical perspective your arguments against it are lacking.


And how familiar are you with modern mathematics?

I am sure I am not as familiar as you are. However, that is beside the point. The issue concerning the transfinite is hotly debated even amongst modern day mathematicians. S. Feferman is an outstanding logician and an expert in the foundations of Mathematics. In his book “In the Light of Logic” (Oxford University Press 1998) he writes…


…there are still a number of thinkers on the subject (Cantor’s transfinite ideas) who in continuation of Kronecker’s attack object to the panoply of transfinite set theory in mathematics…In particular, these opposing points of view reject the assumption of the actual infinite (at least in non-denumerable forms).


If you haven't guessed by know, I am not terribly interested in philosophical hand-waving and hazy prose.

Yes, I can see that the philosophical issues involved are not your strong point. You can dismiss it by name calling, but that does not make it go away. As to whether or not uncountable sets do exist, we will see.

Sincerely,

Brian

[Rand]

Like I said in a previous post, I made a mistake in replying to the parts of this thread that are not directly related to the fact being discussed (the uncountability of R). So, with that in mind, I will ignore all of your last post except one small part, which I only reply to because it questions something very important to me.


Today @ 02:28 AM post located here (http://www.theologyweb.com/forum/showthread.php?s=&postid=135657#post135657)
Brian:

Rand, I certainly could have misconstrued what you were saying. You said…

...

Would you mind explaining to me what this “strict view” entails?


In the middle was a statement I made about God's omnipotence. At times, people (wrongly) assume that "omnipotence" means "the power to do anything" without the provision that the "anything" must be possible. I quoted Dr. Ott to reinforce this point, but perhaps I should elaborate.

It is impossible to make (or conceive of) a square circle, because the definitions of "square" and "circle" are contradictory. So, if someone includes under the heading of "omnipotent" the ability to make square-circles, then God is not omnipotent, because God cannot make a square circle. However, God is omnipotent when understood as Dr. Ott explained in the book I quoted earlier.

Hope I explained that well enough, and I hope we can now get back to discussing mathematics.

[Brian]

Hello Rand!

Thank you for the explanation of your view of omnipotence. It certainly seems orthodox. Please forgive me for my mistaken charge. I have one question, though. How does HRG’s proof assert that God’s omnipotence would include God being able to do something contrary to His nature (for instance lying), or violating the law of non-contradiction? HRG’s assertion was: Let A be the set of all actions that God can take. Since it seems as if HGR’s assertion does not necessarily include violations of logic or God’s character, I still do not understand why you would say, “With such a strict view of omnipotence, I would have to say that God is not omnipotent.” Could you clarify this for me? (Parenthetically, this discrepancy was why I thought you were not orthodox in your view of omnipotence.)


No-- I only want to prove one thing. I want to show that if you have a set A and its power set, P(A), then any function g that maps A to P(A) is not onto.

That is fine. However, if you go back and look you will see your initial intention was to prove that “no set is equivalent to its power set.” But let’s start here and then move forward. I am sure you will agree that there exists an “onto” mapping from P(A) to A since A is a subset of P(A). So when you assume that g is a “onto” mapping from A to P(A), then you are assuming that there is a bijection between the two sets (Schroeder-Bernstein Theorem). When you prove that there is no such g, then you are proving that no such bijection exists. I have already asked this question once (maybe in a slightly different form), but how do you get from “there exists no bijection between A and P(A)” to “|A|=/=|P(A)|”? You really should not object to my wording, but if you do then let me rephrase the question: how do you get from “there exists no onto mapping from A onto P(A)” to “|A|=/=|P(A)|”?

This question is at the root of my issues with SD, and more specifically DOU. DOU assumes that if two infinite sets have the same cardinality, then there exists a bijection between them. Or if you want to phrase it differently, if two infinite sets have the same cardinality, then both sets map “onto” each other. As I have already asserted, there is no proof demonstrating this. My suspicion is that the way you get to |A|=/=|P(A)| is by way of this definition.


It is not necessary that I prove a definition-- that is why it is called a definition. That is the way mathematicians define "countable." From a very naive view, we only need the definition of "cardinal number" to understand what countable means.

First off, all definitions must be consistent with itself, and other definition and theorems within the system. Just because someone proposes a definition, does not make it valid. SD has never been shown to be consistent with, for example, Cantor’s Countable Union Theorem. There is no proof of the absolute consistency of the DOC and DOU with respect to all mapping techniques and all possible infinite sets. While it is obvious to me that if two infinite sets are in bijection they have the same cardinal number, it is not so obvious that if a bijection does not exist between two infinite sets that they have different cardinalities.


Sorry-- you have not presented a counterexample. Look in your list again; where is the set of all prime numbers?

I am a little confused by this. Here is my list again P(P): {0, {2}, {3}, {2,3}, {5}, {2,5}, {3,5}, {2,3,5}, {7}, {2,7}, {3,7}, {2,3,7}, {5,7}, {2,5,7}, {3,5,7}, {2,3,5,7}, etc…}. The set of all prime numbers is contained in the ellipsis. If you want I could list it this way P(P): {0, {2}, {3}, {2,3}, {5}, {2,5}, {3,5}, {2,3,5}, {7}, {2,7}, {3,7}, {2,3,7}, {5,7}, {2,5,7}, {3,5,7}, {2,3,5,7}, …{p1, p2, p3, …, pn}, …}. I have indicated how every element of P(P) would be in the list when the work was done. (If you are not sure of the pattern I will be happy to explain, but I suspect that is not the issue.) I have also shown how this list can be put into a one-to-one correspondence with N. Could you please be more specific with your objection to my counter example? Thank you for the effort you are putting into this discussion.

Sincerely,

Brian

[Rand]

I do not want to go into the HRG's argument again, so I will skip that part...


Today @ 04:05 AM post located here (http://www.theologyweb.com/forum/showthread.php?s=&postid=135685#post135685)
Brian:

That is fine. However, if you go back and look you will see your initial intention was to prove that “no set is equivalent to its power set.” But let’s start here and then move forward. I am sure you will agree that there exists an “onto” mapping from P(A) to A since A is a subset of P(A). So when you assume that g is a “onto” mapping from A to P(A), then you are assuming that there is a bijection between the two sets (Schroeder-Bernstein Theorem). When you prove that there is no such g, then you are proving that no such bijection exists. I have already asked this question once (maybe in a slightly different form), but how do you get from “there exists no bijection between A and P(A)” to “|A|=/=|P(A)|”? You really should not object to my wording, but if you do then let me rephrase the question: how do you get from “there exists no onto mapping from A onto P(A)” to “|A|=/=|P(A)|”?

(As a side note, I did not assume a bijection existed between A and P(A). I simply assumed the function was onto, but that does not matter for our discussion.)

This is the part where we run into trouble. Since you do not have a mathematics background, we cannot get into a nitty-gritty axiomatic development of set theory, where cardinal numbers are defined in terms of ordinal numbers (and ordinal numbers are epsilon-images of sets). However, we do not have to go there. Below I will trace a development of cardinal numbers that one might see in a naive introduction to set theory. This should suffice to prove my point.

You keep insisting on a difference between "no bijection exists between two sets" and "the cardinal number of two sets is different," when there is no difference. We can forego the nasty definitions and develop something like the following (I have done this once before, but I will try to explain it in more detail):

Definition: Two sets are said to be equivalent if there exists a bijection between them.

Definition: The cardinal number of a set is an arbitrary representative from the class of all sets equivalent to it.

So, say we have two sets, A and B. Let's also assume that no bijection exists between them. Thus, they belong to two disjoint classes of mutually equivalent sets-- in other words, we have the class of sets that are all equivalent to A and the class of sets that are all equivalent to B. Furthermore, these classes are disjoint (from the transitive nature of set equivalence).

Lets keep going with the same example. The cardinal number of A is an arbitrary representative from the class described above (we will denote it a). Similarly, the cardinal number of B is b. Note that these "numbers" are actually sets. Since the two classes were disjoint, a and b cannot be the same number (set). So, we can go from "a bijection does not exist between A and B" to "A and B have different cardinal numbers."

Your other questions seemed to stem from this one... Hopefully you now see that there is no difference between sets being equivalent and having the same cardinal number.


I am a little confused by this. Here is my list again P(P): {0, {2}, {3}, {2,3}, {5}, {2,5}, {3,5}, {2,3,5}, {7}, {2,7}, {3,7}, {2,3,7}, {5,7}, {2,5,7}, {3,5,7}, {2,3,5,7}, etc…}. The set of all prime numbers is contained in the ellipsis. If you want I could list it this way P(P): {0, {2}, {3}, {2,3}, {5}, {2,5}, {3,5}, {2,3,5}, {7}, {2,7}, {3,7}, {2,3,7}, {5,7}, {2,5,7}, {3,5,7}, {2,3,5,7}, …{p1, p2, p3, …, pn}, …}. I have indicated how every element of P(P) would be in the list when the work was done. (If you are not sure of the pattern I will be happy to explain, but I suspect that is not the issue.) I have also shown how this list can be put into a one-to-one correspondence with N. Could you please be more specific with your objection to my counter example? Thank you for the effort you are putting into this discussion.

Sorry-- the counter-example just does not work (though I can understand why you think it does). Look at your sets more closely:
0 (a finite set)
{2} (a finite set)
{3} (a finite set)
{2,3} (a finite set)
{5} (a finite set)
{2,5} (a finite set)
{3,5} (a finite set)
{2,3,5} (a finite set)
{7} (a finite set)
{2,7} (a finite set)
{3,7} (a finite set)
{2,3,7} (a finite set)
{5,7} (a finite set)
{2,5,7} (a finite set)
{3,5,7} (a finite set)
{2,3,5,7} (a finite set)
.
.
.

Although your list contains an infinite number of sets, it does not contain any set with an infinite number of elements. You simply build this finite sets up, but you never make that critical jump to the set of all primes. Think of it this way:

Your first set has zero elements, your second set has one element, your third set has one element, and your fourth set has two elements. Now, I don't want to deduce a rule for the number of elements in one of your sets, so lets just pretend that the sets in your list increase by one each time (although this is not exactly the case, it corresponds to the scenario well enough). Now look:

Set#1 - 1 element
Set#2 - 2 elements
Set#3 - 3 elements
.
.
.

Now, the sets I have described "behave" much like the list of sets that you gave me, but in my example, it is easier to see that no set in my list contains an infinite number of elements. Because if you say, "Rand, go to set #n," you have already told me that it is a finite set. Similarly, if I was able to find a rule describing the number of elements in the sets of your list as the appear, if I said, "Brian, go to set #n," you would be able to tell me exactly how many element are in that set. So, you only list the finite subsets of the prime numbers, and thus, your counter-example does not work.

Well, you certainly could put your example into a 1-1 correspondence with N, but unfortunately, it just wasn't P(P). :)

[Brian]

Hello Rand!


As a side note, I did not assume a bijection existed between A and P(A). I simply assumed the function was onto…

You are the mathematician here, not me. Surely you understand how your assumption that there exists an onto mapping from A onto P(A) is the same as assuming a bijection between A and P(A)? If you would like me to elaborate I would be happy to do so.


Since you do not have a mathematics background, we cannot get into a nitty-gritty axiomatic development of set theory, where cardinal numbers are defined in terms of ordinal numbers (and ordinal numbers are epsilon-images of sets). However, we do not have to go there.

It is interesting to note that both Cantor’s proof for the uncountability of R, and his Power Set Theorem were formulated in the late 1800’s, which was prior to the development of 20th century axiomatic set theory.


You keep insisting on a difference between "no bijection exists between two sets" and "the cardinal number of two sets is different," when there is no difference.

To be more accurate, you assume that they are the same. There has been no proof showing that the set of all sets whose cardinality is the same and the set of all sets where a bijection exists is the same set! I am just pointing out the assumption you are making. By the way, this becomes a second assumption made in your proof, which violates the rules of RAA.


Definition: Two sets are said to be equivalent if there exists a bijection between them.

I am not sue what you mean by the term equivalent. Does this mean the two sets have the same cardinality? If so, what do you mean by cardinality? Here is my definition…

Definition: The cardinality of a set is the number of members it contains.

If you are saying, “two sets have the same cardinality if there exists a bijection between them,” then I have another question. If this is a definition, then shouldn’t it be an “if and only if” statement? If this is your intention, then I say that this definition is no different from SD and suffers with the same problems. However, I do agree that if there exists a bijection between two sets, then they have the same cardinality. What I don’t agree to is that if two sets have the same cardinality, then there exists a bijection between them.


Definition: The cardinal number of a set is an arbitrary representative from the class of all sets equivalent to it.

Is this the same as the definition I put forward? Would this be the definition Cantor used when he put forth his theorems? If not, I would rather use my definition. I certainly can provide you with references for the definition.


So, say we have two sets, A and B. Let's also assume that no bijection exists between them. Thus, they belong to two disjoint classes of mutually equivalent sets-- in other words, we have the class of sets that are all equivalent to A and the class of sets that are all equivalent to B.

I suspect this completely begs the question, but will wait for further clarification from you on the definitions being used.

In terms of my counter example, it seems that your objection to my listing was that it would never contain the individual set of all primes, which we have agreed is infinite. Using your simplified algorithm we have:

1 P(1)
2 P(1), P(2)
3 P(1), P(2), P(3)
.
.
.
n P(1), P(2), P(3), …, P(n)
.
.
.
Are you are saying that this algorithm cannot be used to define the infinite set of all primes?

Sincerely,

Brian

[Rand]

Today @ 05:16 PM post located here (http://www.theologyweb.com/forum/showthread.php?s=&postid=135969#post135969)
Brian:

It is interesting to note that both Cantor’s proof for the uncountability of R, and his Power Set Theorem were formulated in the late 1800’s, which was prior to the development of 20th century axiomatic set theory.

Yes, they were, and they were amazing breakthroughs at the time. Unfortunately, a naive view of set theory could cause some problems (Russell's paradox), and later mathematicians had to put set theory on an axiomatic basis to clear up those errors. Fortunately, after set theory was set on a base of axioms, it still had the nice properties that Cantor proved (uncountability of R, etc).


I am not sue what you mean by the term equivalent. Does this mean the two sets have the same cardinality? If so, what do you mean by cardinality? Here is my definition…

Definition: The cardinality of a set is the number of members it contains.

Your definition of carindality is not good, although it gives us a (very) informal view of what cardinality is.

Equivalent means just what I said it means, and this definition (along with a proper definition of cardinal number) gives us the fact that two equivalent sets have the same cardinal number (and two sets with different cardinal numbers are not equivalent).


What I don’t agree to is that if two sets have the same cardinality, then there exists a bijection between them.

That is because you do not know what cardinal numbers are...


Is this the same as the definition I put forward? Would this be the definition Cantor used when he put forth his theorems? If not, I would rather use my definition. I certainly can provide you with references for the definition.

I do not know what definition Cantor used. However, this is a moot point since set theory has developed quite a bit since Cantor. And no, we cannot use your definition, because your definition is not well defined (what does number mean in your definition?). Your definition is only a way to grasp what the concept of a cardinal number is. It does not tell us very much about cardinal numbers themselves, though.


In terms of my counter example, it seems that your objection to my listing was that it would never contain the individual set of all primes, which we have agreed is infinite. Using your simplified algorithm we have:

1 P(1)
2 P(1), P(2)
3 P(1), P(2), P(3)
.
.
.
n P(1), P(2), P(3), …, P(n)
.
.
.
Are you are saying that this algorithm cannot be used to define the infinite set of all primes?

That is exactly what I am saying. Each set you describe has a finite number of elements. Let n be a natural number. There is a set in your list with n elements, no matter what n is. So you have described sets with millions, billions, or trillions of elements, but you never listed an infinite set.

[Brian]

Hello Rand!


Your definition of carindality is not good, although it gives us a (very) informal view of what cardinality is.

Help me out here Rand. I can list numerous sources that will give the same definition for cardinality. Also, I am almost positive that my definition was what Cantor would have been operating under. Since that is what the issue concerns (i.e. Cantor’s proof), then it seems only reasonable to use this widely accepted definition.


Equivalent means just what I said it means, and this definition (along with a proper definition of cardinal number) gives us the fact that two equivalent sets have the same cardinal number (and two sets with different cardinal numbers are not equivalent).

OK. Then I say your definition for equivalence has the same problem as the standard definition for countability.


That is because you do not know what cardinal numbers are...

My understanding of cardinal numbers is that they are numerical representations of the number of elements contained in a set. I also submit that this is what Cantor believed.


I do not know what definition Cantor used. However, this is a moot point since set theory has developed quite a bit since Cantor.

This is just wrong. What definition Cantor used is precisely the point. Cantor’s proof for the uncountability of R is the basis for transfinite mathematics. Whatever developments occurred afterward are developments that depend on Cantor’s theorems.


And no, we cannot use your definition, because your definition is not well defined (what does number mean in your definition?). Your definition is only a way to grasp what the concept of a cardinal number is. It does not tell us very much about cardinal numbers themselves, though.

Number means quantity. Rand, are you interested in defending Cantor or not? If so, then I am going to ask you to defend him on his playground. If you are not willing to do this, then tell me where Cantor was wrong, which will necessarily make my point.


That is exactly what I am saying. Each set you describe has a finite number of elements. Let n be a natural number. There is a set in your list with n elements, no matter what n is. So you have described sets with millions, billions, or trillions of elements, but you never listed an infinite set.

OK. Then tell me how Cantor’s digitalization mapping was accomplished? He uses an algorithm to construct his “new number” that is not unlike the algorithm you said could never construct the set of prime numbers.

Sincerely,

Brian

[Rand]

Today @ 08:12 PM post located here (http://www.theologyweb.com/forum/showthread.php?s=&postid=136055#post136055)
Brian:
Help me out here Rand. I can list numerous sources that will give the same definition for cardinality. Also, I am almost positive that my definition was what Cantor would have been operating under. Since that is what the issue concerns (i.e. Cantor’s proof), then it seems only reasonable to use this widely accepted definition.

The definition you gave allows someone who does not have a mathematics background to understand what cardinal numbers are. However, it is far from precise. If you want to quote any set theory book used by college students that defines "cardinal number" as "the number of elements in a set," be my guest.


OK. Then I say your definition for equivalence has the same problem as the standard definition for countability.

You are free to show where a problem might exist, but I assure you that there is none. I outlined this previously, so I will not do so again.


My understanding of cardinal numbers is that they are numerical representations of the number of elements contained in a set. I also submit that this is what Cantor believed.

Carindal numbers act as "numerical" representations of the number of elements in a set (ignoring the fact that infinite sets are not represented with numbers). However, what cardinal numbers are is different. This can be seen in my definition of cardinal number.


This is just wrong. What definition Cantor used is precisely the point. Cantor’s proof for the uncountability of R is the basis for transfinite mathematics. Whatever developments occurred afterward are developments that depend on Cantor’s theorems.

I do not care what Cantor may or may not have used as the definition of carindal number. Your insistence on using his definition (assuming it is different) is ludicrous. Let's look at an analogy:

Person A posts on a message board saying that gravity does not work the way Isaac Newton thought, and he can prove this based on the planetary motion of Mercury.

Person B responds by saying that Mercury's motion can be explained better using Einstein's theory of general relativity. Of course, Einstein's theory does explain the motion of Mercury to a degree of accuracy that Newton's methods just do not approach.

At this point, the debate is over. Person B has won, because new developments change the information we have available.

So I once again state that Cantor's definition (which you have still not provided) does not matter. The definition given by mathematicians matters. When I say "cardinal number," I mean something very precise. When you say "cardinal number," you mean something that is similar to the actual definition, but it is not precise.

If you want to speak about something in the world of mathematics, it is only reasonable that you use the definitions that are accurate, and your definition of cardinal number is not accurate; once again, that definition of "cardinal number" can only be used to give someone a vague idea of what the concept "cardinal number" is all about.


Number means quantity. Rand, are you interested in defending Cantor or not? If so, then I am going to ask you to defend him on his playground. If you are not willing to do this, then tell me where Cantor was wrong, which will necessarily make my point.

I am not interested in defending Cantor at all-- I am interested in presenting basic mathematics. You are making claims about set theory, and set theory does not rest solely upon the work of Georg Cantor.


OK. Then tell me how Cantor’s digitalization mapping was accomplished? He uses an algorithm to construct his “new number” that is not unlike the algorithm you said could never construct the set of prime numbers.

Do you still not understand why your counterexample does not work? You will never construct the set of all primes (but you do construct an infinite list of sets).

Similarly, Cantor's argument constructs an infinite list of numbers, just as you constructed an infinite list of sets (and when these numbers are viewed as the decimal representation of a number in the continuum, the result follows).

I just realized that Cantor's argument has never been presented in this thread, so for any lurkers present, here we go:

***********

This argument was proposed by Georg Cantor in the late nineteenth century to show that the real numbers are not countable. We will do this by showing that the open interval (0,1) is itself uncountable. We prove this by contradiction.

Assume that (0,1) is countable. Since it is countable, it can be put into a one-to-one correspondence with the natural numbers. Now, we will list this one-to-one correspondence; note that each number in the interval (0,1) can be viewed using its decimal representation:

1 - .(a1,1)(a1,2)(a1,3)(a1,4)...
2 - .(a2,1)(a2,2)(a2,3)(a2,4)...
3 - .(a3,1)(a3,2)(a3,3)(a3,4)...
4 - .(a4,1)(a4,2)(a4,3)(a4,4)...
.
.
.

The notation that I was forced to use is pretty terrible, but the two numbers following "a" in the parentheses is the place held by each digit (viewed like a matrix would be). So (a20,53) refers to the 51st digit in the 20th number.

Now, we will show that there is a number in the open interval (0,1) that has not been mapped to. This number can be viewed with a decimal representation as:

.(b1)(b2)(b3)(b4)...

Now, to define (b1) we will look at (a1,1), and to define (b2) we will look at (a2,2), and so on. That is why this method of proof is called "diagonalization."

If (a1,1) = 4, we say (b1) = 7.
If (a1,1) <> 4, we say (b1) = 4.

Do this for every (bn), and we have contructed a number that was not mapped to, because it differs from each number listed in the (an,n) digit. Thus, (0,1) is not countable.

***********

OK, now back to Brian-- you ignore the fact that my argument for the power set theorem does not rest upon cardinal numbers at all...

[Brian]

Hello Rand!

Here are the two definitions you have presented…

Definition: Two sets are said to be equivalent if there exists a bijection between them.

Definition: The cardinal number of a set is an arbitrary representative from the class of all sets equivalent to it.

You claim that the cardinal number of a set is a representation of all sets where a bijection exists. I reject this on the basis that it has never been proved that two sets which have the same cardinality must have a bijection. Your definition of cardinality completely begs the question.

Here are a few sources that define cardinality in terms of “number of elements in a set”:

http://www.earlham.edu/~peters/writing/infapp.htm
http://whatis.techtarget.com/definition/0%2C%2Csid9_gci498885%2C00.html
http://planetmath.org/?op=getobj&from=objects&name=Cardinality
http://eksl-www.cs.umass.edu/~bburns/cs250/Lecture26.ppt
http://plato.stanford.edu/entries/set-theory/primer.html#5
http://www.math.utep.edu/Faculty/valdez/notes2303.html

I am sure that I can list many more, but I think you get the point. My definition is common, and is something that I agree to. What you have done with your definition of cardinality is defined it in terms of bijection. You completely ignore the possibility that two infinite sets might have the same number of elements and yet a bijection does not exist between them. For instance, what would happen if you had an infinite set that is made up of 2 disjoint countable infinite sets? By definition this set would not have a bijection with N, yet by Cantor’s Countable Union Theorem it would be countable! I am looking at a paper written by Dr. Webster Kehr titled Hinged Sets and the Answer to the Continuum Hypothesis. In the paper he claims to prove R is countable by this very method. If you want to reach him you can find him @ webster.r.kehr@mail.sprint.com. Also, I have been in correspondence with Dr. Alexander Zenkin of the Russian Academy of Sciences. He is a doctor of Physical and Mathematical sciences, and is the leading research scientist for the computing center there. His web site is :www.com2com.ru/alexzen. He too raises the same issues with Cantor and meta-mathematics. (Dr. Zenkin has been most gracious to personally answer many of the questions I have had concerning this topic.)


I am not interested in defending Cantor at all-- I am interested in presenting basic mathematics. You are making claims about set theory, and set theory does not rest solely upon the work of Georg Cantor.

The uncountability of R rests solely on Cantor.


Similarly, Cantor's argument constructs an infinite list of numbers, just as you constructed an infinite list of sets (and when these numbers are viewed as the decimal representation of a number in the continuum, the result follows).

It is you that is missing the point. How does Cantor “build” his diagonal number? One digit at a time. Using your representation and the algorithm you put forth it looks this way:

1 b(1)
2 b(1) b(2)
3 b(1) b(2) b(3)
.
.
.
n b(1) b(2) b(3)…n
.
.
.
Now allow me to quote what you say I cannot do…


1 P(1)
2 P(1), P(2)
3 P(1), P(2), P(3)
.
.
.
n P(1), P(2), P(3), …, P(n)
.
.
.

How is this any different from what Cantor did? As I see it you have two choices: (1) If Cantor’s algorithm actually does create an infinite string of digits, i.e., his new number, then my counter example is established. (2) If Cantor’s algorithm is not valid, then there is no proof that R is uncountable.


OK, now back to Brian-- you ignore the fact that my argument for the power set theorem does not rest upon cardinal numbers at all...

Your argument for the power set theorem rests upon a second assumption as noted several times.

Sincerely,

Brian

[Rand]

You claim that the cardinal number of a set is a representation of all sets where a bijection exists. I reject this on the basis that it has never been proved that two sets which have the same cardinality must have a bijection. Your definition of cardinality completely begs the question.

Once more-- cardinal numbers are defined so that sets have the same cardinality only when there is a bijection between them. Your mistaken idea of the nature of cardinal numbers is causing you quite a bit of confusion.

At the bottom of this post, I will show you what a cardinal number actually is from a few real sources. But first, let's examine your links:

http://www.earlham.edu/~peters/writing/infapp.htm

This place introduces cardinal numbers with the definition you want. Unfortunately, this website is designed to be a (very) naive introduction to set theory. They are trying to present the material in a way that would allow someone with no mathematical training (like yourself) to understand. Of course, later in the article, they say the following, "Two sets have the same cardinality iff they can be put into one-to-one correspondence." They say this as a definition-- not a theorem, which should reinforce the fact that this website is only an extremely basic introduction.

http://whatis.techtarget.com/definition/0%2C%2Csid9_gci498885%2C00.html

Another site for very basic (non-rigorous) introductions and definitions.

http://planetmath.org/?op=getobj&amp;from=objects&amp;name=Cardinality

Funny-- this is the definition I saw on this site: "Sets A and B have the same cardinality if there is a one-to-one and onto function f from A to B."

http://eksl-www.cs.umass.edu/~bburns/cs250/Lecture26.ppt

This is another very basic introduction to set theory that is for a computer science class... Not exactly the height of mathematical sophistication.

http://plato.stanford.edu/entries/set-theory/primer.html#5

They did not give your definition (that I saw, anyway). They did give this, though: "Sets A and B have the same cardinality if there is a one-to-one function f with domain A and range B." This means that the cardinality of two sets is the same exactly when a bijection exists between them.

http://www.math.utep.edu/Faculty/valdez/notes2303.html

This was a broken link.


I am sure that I can list many more, but I think you get the point. My definition is common, and is something that I agree to.

The definition you gave is nice-- for people with little to no mathematical knowledge. However, if you want to argue about very precise mathematical theorems, you need to use precise definitions.


For instance, what would happen if you had an infinite set that is made up of 2 disjoint countable infinite sets? By definition this set would not have a bijection with N, yet by Cantor’s Countable Union Theorem it would be countable!

Of course there exists a bijection between the union of two countably infinite sets and the natural numbers, and I can prove it:

Let A and B be two countable infinite sets, where
A={a1,a2,a3,a4,...} B={b1,b2,b3,...}Define a function, g from the natural numbers to A+B as follows:
g(1) = a1
g(2) = b1
g(3) = a2
g(4) = b2
g(5) = a3
g(6) = b3
.
.
.



It is you that is missing the point. How does Cantor “build” his diagonal number? One digit at a time. Using your representation and the algorithm you put forth it looks this way:

<snip>

How is this any different from what Cantor did? As I see it you have two choices: (1) If Cantor’s algorithm actually does create an infinite string of digits, i.e., his new number, then my counter example is established. (2) If Cantor’s algorithm is not valid, then there is no proof that R is uncountable.

Tell me this-- in your list, where is the set of all primes? It is *not* defined in your algorithm, since each step you take defined a finite set. Once again, your list is infinite (just like Cantor's), but the sets in your infinite list are not.

Finally, here are three presentations of cardinal numbers from set theory books that might actually be used to study set theory.

Definition: By a cardinal number of a power m we mean an arbitrary representative M of a class of mutually equivalent sets.

You can find this definition in E. Kamke's Theory of Sets, Dover Publications 1950, page 18.

Definition: The transfinite recursion theorem presents us with a unique function E with domain A such that for any t in A:
E(t) = {E(x)| x<t}
Let a = ran(E); we will call a the epsilon-image of the well-ordered structure (A,<).

Definition: Let < be a well-ordering on A. The ordinal number of (A,<) is its epsilon image.

Definition: For any set A, define the cardinal number of A to be the least ordinal equinumerous <equivalent> to A.

These definitions can be found in Herbert B. Enderton's book, Elements of Set Theory, Academic Press 1977, pages 182, 189, 197.

Definition: The Frege-Russell definition of cardinal numbers is beautiful in its simplicity. The cardinal number of the set A is the class of all sets equipollent (equivalent) to A.

This definition can be found in Axiomatic Set Theory by Patrick Suppes, Dover Publications 1972, page 109.

So-- if you agree to one of the definitions above (I prefer Kamke's or Suppe's), can we continue?

[Brian]

Hello Rand!


Once more-- cardinal numbers are defined so that sets have the same cardinality only when there is a bijection between them. Your mistaken idea of the nature of cardinal numbers is causing you quite a bit of confusion.

What is the cardinality of the following set: {1, 2, apple, *}? Please be sure to tell me how you know.


At the bottom of this post, I will show you what a cardinal number actually is from a few real sources…This place introduces cardinal numbers with the definition you want. Unfortunately, this website is designed to be a (very) naive introduction to set theory. They are trying to present the material in a way that would allow someone with no mathematical training (like yourself) to understand.

A few “real” sources? You do not consider Dr. Peter Suber to be a real source? What is your basis for that? Also, just because the paper is a “very naïve introduction to set theory” that does not mean the definitions used are not valid. Allow me to quote the bottom of his paper.


Bibliographic note. Most of the theorems and proofs in this crash course were discovered by Georg Cantor (1845-1918) and published in a series of monographs starting in 1870. He published two summary statements of his results in 1895 and 1897, which have been translated into English by Philip E. B. Jourdain as Contributions to the Founding of the Theory of Transfinite Numbers, Dover Publications, 1955. My exposition of Cantor's results is indebted to three more recent authors: Stephen Cole Kleene, Introduction to Metamathematics, North-Holland Pub. Co., 1952; Abraham Fraenkel, Abstract Set Theory, North-Holland Pub. Co., 1953; and Geoffrey Hunter, Metalogic, University of California Press, 1971.

Again, Cantor and his work is the foundation for Transfinite number theory. The uncountibility of R is based solely on his proof. The definitions that he used are valid. Cardinality is the number of elements in a set. By the way, none of the links were broken, and you need to go back and read more carefully. All of these sources understand cardinality in terms of the number of elements in a set. I will be happy to quote each portion if you are still having trouble finding it.


The definition you gave is nice-- for people with little to no mathematical knowledge. However, if you want to argue about very precise mathematical theorems, you need to use precise definitions.

Why don’t you tell that to Cantor.


Tell me this-- in your list, where is the set of all primes? It is *not* defined in your algorithm, since each step you take defined a finite set. Once again, your list is infinite (just like Cantor's), but the sets in your infinite list are not.

Do you understand the diagonal procedure? Do you understand how Cantor creates his diagonal number, and how he makes sure that it is compared to every element in the listing? This response leaves me with some doubt. The set of elements that makes up Cantor’s new number are the digits b1, b2, b3, … At each step in the algorithm a digit is added, but it is always finite, i.e. not real. To ask your question, at what point does this number become real?

Brian

[Rand]

Today @ 03:44 AM post located here (http://www.theologyweb.com/forum/showthread.php?s=&postid=136388#post136388)
Brian:
What is the cardinality of the following set: {1, 2, apple, *}? Please be sure to tell me how you know.

The cardinality of that set is 4, because it is equivalent to the set {{},{{}},{{},{{}}},{{},{{}},{{},{{}}}}. We name this set the natural number "4," so the cardinality of your set is 4.


<snip>

"Rand-- make your argument without relying on the definitions of the terms involved. Instead, use these notions that are not actually definitions."

No thanks, Brian. I've given reference to three common sources of set theory (one of which quoted Russell and Frege) for the definition of what a "cardinal number" is. If you don't accept one of these definitions, then you are not talking about set theory, but something else.

Cantor's original work, while excellent, was not consistent. Bertrand Russell found his paradox, and mathematicians spent a while ironing things out. The definition you give cannot be used for our purposes because it is not well defined. Your definition, if was:

The cardinal number of a set is the number of elements in the set.

This definition cannot be used, because set theory does not assume a priori the existence of numbers! In set theory, we have no idea what numbers are until we define the natural numbers (using a successor operation on the null set). Then we define the rational numbers (in terms of the natural numbers). Then we define the irrational numbers (using Dedekind cuts or Cauchy sequences of rational numbers). Finally, we show that this wonderful structure satisfies the axioms of a peano system (we can do this at each step along the way, actually); only then do we have the real numbers.

By the way, I have not been able to find Cantor's original work to see what he used as a definition of "cardinal number." The sources I have seen make it seem like he used a definition not unlike one that I gave, although I cannot be sure. Perhaps HRG can tell us is he knows... He may have seen Cantor's original work in his mathematical studies. Since you have been so busy telling me what Cantor said, perhaps you can quote his work?

You are mistaking the concept of cardinality with the actuality of a cardinal number. You said, "Cardinality is the number of elements in a set." This is true (informally), but "cardinality" is a concept. A "cardinal number" is a set (or class) that has a definition. Your sources understand cardinal numbers as "things" that tell us how many elements are in a set. This is a very good, informal view, but those "things" have a very precise definition, and until you are willing to accept that, you will not be able to study set theory.


Do you understand the diagonal procedure? Do you understand how Cantor creates his diagonal number, and how he makes sure that it is compared to every element in the listing? This response leaves me with some doubt. The set of elements that makes up Cantor’s new number are the digits b1, b2, b3, … At each step in the algorithm a digit is added, but it is always finite, i.e. not real. To ask your question, at what point does this number become real?

I fully understand the diagonal argument. You are the one who (obviously) does not understand it, and this is plain from the simple fact that you cannot tell me where the set of all primes is in your list.

Your method and Cantor's method are the same-- we allow the 'algorithm' to exhaust itself (a countable number of times). After this, you have an infinite list of sets. This list of sets contains every finite subset of the prime numbers. After his method, Cantor was left with an infinite list of digits. These digits form one real number. This was useful, since it proves that the real numbers are uncountable. Your list (while infinite) was not very useful, since it left out quite a few subsets of the prime numbers (namely all of the infinite ones).

I am really getting tired of this conversation... I've proven the uncountability of R three different ways (Cantor's proof, the power set theorem, and the measure theory proof), so there is very little else to talk about, until you learn the definitions of these terms. I would suggest Kamke's book-- it is not an axiomatic presentation of set theory, so it is accessible to non-mathematicians, and it is cheap, from Dover Publications.

So, to sum it all up for the kiddies watching at home:

Cantor's Proof:
The real numbers are not countable, because if they were, there would exist a bijection between R and N(1). No such bijection exists (by way of contradiction), so the cardinal number(2) of R is greater than the cardinal number of N. Thus, from the definition of countable(3), R is not countable.

Measure Theory Proof:
All countable sets have measure 0.
The measure of R is infinite.
Thus, R is not countable.
(look kids, no resorting to cardinal numbers here!)

Power Set Theorem:
For any set A, its power set has a greater cardinal number, because if the cardinal numbers were the same, then there would exist an onto function between A and P(A). However, no function between A and P(A) is onto, so the two sets are not equivalent. Thus, the cardinal number of P(A) is greater than the cardinal number of A.

(1)Two sets are equivalent if a bijection exists between them.

(2)The cardinal number of a set is an arbitrary representative from the class of sets equivalent to it.

(3)A set is countable if its cardinal number is less than or equal to the cardinal number of the natural numbers.

(4)The cardinal number m is less than the cardinal number n iff M is equivalent to a proper subset of N, but N is equivalent to no subset of M, where M and N are disjoint representatives of the cardinal numbers m and n, respectively.

Have fun.

[Brian]

Hello Rand!

In response to my question of the cardinality of the following set: {1, 2, apple, *}, you answered…


The cardinality of that set is 4, because it is equivalent to the set {{},{{}},{{},{{}}},{{},{{}},{{},{{}}}}.

Please be patient with me, but I do not understand how {1, 2, apple, *} is equivalent to {{},{{}},{{},{{}}},{{},{{}},{{},{{}}}}. Can you explain that to me? Specifically, how are the elements of apple and * notated, and why that particular notation? But if you don’t mind, please label everything for me.


"Rand-- make your argument without relying on the definitions of the terms involved. Instead, use these notions that are not actually definitions."

This is a gross misrepresentation, and unfair. I will try and never misrepresent your position. Please give me the same courtesy.


Cantor's original work, while excellent, was not consistent.

What is the nature of Cantor’s inconsistency? Was it his understanding of cardinality?


By the way, I have not been able to find Cantor's original work to see what he used as a definition of "cardinal number."

Did you note the bibliographical information at the end of Suber’s paper? I quoted it for you. He said that Cantor discovered most of the definitions he proposed. The works were also listed. I am sure you can look them up. By the way, did you go back and re- check the other references? You did not mention it.


I fully understand the diagonal argument. You are the one who (obviously) does not understand it, and this is plain from the simple fact that you cannot tell me where the set of all primes is in your list.

Hmmmm. Firstly, why would me not being able to tell you where the set of all primes is (which I have, you just don't like my answer) have anything to do with my understanding of Cantor's diagonal proof? Let’s see if I do not understand it. The DT (diagonal theorem) starts out with the assumption of “given any countable listing of R(0,1).” It then places N in bijection with this set. As I have already mentioned, the justification for doing this is the contrapositive of DOU, which states that if a set is countable it can be placed into bijection with N.

At this point DT asks, “Can every element of R(0,1) be in this countable listing?” To answer this question the DT creates an element of R(0,1), the NN (new number), which is not mapped to by any element of N. It does this with a sequence and algorithm. The algorithm I will use is: “if the nth digit of the nth element of the list is not a ‘5,’ then the nth digit of the NN will be a ‘5,’ otherwise the nth digit of the NN will be a ‘6.’

Once this work is done, DT uses DOU. Since no specific element of N taken from the listing maps to the NN, therefore N cannot map onto all of the elements of the listing, and R(0,1) cannot be placed into bijection with N. Therefore, by using DOU it is concluded that R(0,1) is uncountable. Note: Cantor’s proof relies completely on DOU.

Have I understood this?

Note, the diagonal theorem treats the listing not as a one column, infinite row listing, but rather as a two-dimensional array. I will refer to this as “single set array” (SSA) because it views a single set as a two-dimensional array of characters. Note that a SSA has two different infinite orientations: an infinite column orientation which is by definition in bijection with N, and an infinite row orientation, which according to our original assumption is in bijection with N. Without this character array there would be no such thing as diagonalization. The term “diagonalization” refers to the diagonal cells of a character array, beginning from the top left corner. The algorithm is applied to this character array to build its diagonal number. Diagonalization establishes a “link” between the two sets that make up the columns and the rows. As already mentioned, these two sets are in bijection with N.

This last paragraph is setting up my argument for why Cantor’s construction of the NN, is no different from my listing of P(P). I will continue with it if you are still interested.


I am really getting tired of this conversation. I've proven the uncountability of R three different ways (Cantor's proof, the power set theorem, and the measure theory proof), so there is very little else to talk about, until you learn the definitions of these terms.

First off, there is no one keeping you here. Secondly, you claim to have proven it, and I have stated what some of my objections are. As far as I am concerned, what we are now doing is hammering out your “proofs” relative to my objections. Once this is done, then we can decide whether or not you have proven anything. I will await your comments.

Sincerely,

Brian
(P.S. – I may not be able to respond again until Thursday or Friday, assuming you want to continue.)

[Defenestrator]

Brian:
Please be patient with me, but I do not understand how {1, 2, apple, *} is equivalent to {{},{{}},{{},{{}}},{{},{{}},{{},{{}}}}. Can you explain that to me? Specifically, how are the elements of apple and * notated, and why that particular notation? But if you don’t mind, please label everything for me.


Maybe this will help:

{ {} , {{}} , {{},{{}}} , {{},{{}},{{},{{}}} } }. <== I think he needed one more closing bracket.

1- {}
2- {{}}
apple- { {} , {{}} }
*- { {} , {{}} , {{},{{}}} }

[Brian]

Hello Defenestrator!

Thank you for your help. I understand a little better. Ok, the first element in the set is definied as {}. The second element is definied as {{}}. Whyi sn't the third element be defined as {{{}}}? It seems as if you define the third element as the union of {} and {{}}. Is {{{}}} and {{}, {{}}} the same?

Thanks!

Brian

[Defenestrator]

Here is what I think Rand is doing. I'm going to change notation a little.

Your first symbol "1," Rand defined as {}, which I took to be f, the set with no elements. 1=f

Your second symbol "2," Rand defined as {{}}, which I took to be {f}, a set with one element. 2={f}={1}

Your third symbol "apple," Rand defined as {{}, {{}}}, which I took to be {f,{f}}, a set with two elements. apple={f,{f}}={1,2}

Your fourth symbol "*" under my scheme would be {f, {f}, {f,{f}} }, a set with three elements. *={f, {f}, {f,{f}} }={1,2,apple}

The key is when going from 1 to 2 or from 2 to apple or, in general, from one symbol to the next, is to increase the number of elements in the next set by 1 and to only use elements in that next set that have already been defined.

[Rand]

Yesterday @ 11:16 PM post located here (http://www.theologyweb.com/forum/showthread.php?s=&postid=137024#post137024)
Brian:
Thank you for your help. I understand a little better. Ok, the first element in the set is definied as {}. The second element is definied as {{}}. Whyi sn't the third element be defined as {{{}}}? It seems as if you define the third element as the union of {} and {{}}. Is {{{}}} and {{}, {{}}} the same?

The two sets that you mentioned, {{{}}} & {{},{{}}}, are not the same. The way those sets are constructed happens to be a very efficient way to define the natural numbers-- there are other ways, but that one has some interesting properties. I showed this before when I explained the nature of the elements of the set of natural numbers. Here is a more detailed explanation of why I said the cardinality of the set you described was 4. I set up a bijection between your set and the set I described. Specifically, the mapping I had in mind was:
1 -> {}
2 -> {{}}
apple -> {{},{{}}}
* -> {{},{{}},{{},{{}}}}
Now, I did not know how you defined "apple" or "*," but this did not matter. It was enough that I created a bijection between the two sets. This shows that the sets are equivalent. That showed that the sets have the same cardinal number (because of the definition of cardinal number). The set I described is the natural number "4," and so we call the cardinal number of both sets "4".

This is one reason why it is important to use appropriate definitions for the terms we are working with.


This is a gross misrepresentation, and unfair. I will try and never misrepresent your position. Please give me the same courtesy.

I do not believe that it was a gross misrepresentation. Perhaps it was a bit rude, but quite frankly, it does not seem like you are willing to accept the definitions of the terms we are dealing with.

Until you are willing to accept the following definitions (or one of the others that I proposed), then there is nothing to talk about:

Definition: Two sets are equivalent iff a bijection exists between them.

Definition: The cardinal number of a set is an arbitrary representative from the class of all sets mutually equivalent to it.

Note that the second definition is well defined since set set equivalence acts like an equivalence relation (ie it is reflexive, symmetric, and transitive).

Definition: A set is countable iff its cardinal number is less than or equal to Aleph_0.


What is the nature of Cantor’s inconsistency? Was it his understanding of cardinality?

Neither of us has presented Cantor's definition of "cardinal number." Cantor's almost assuredly understood cardinality to represent the number of elements in a set (the concept of cardinality tells us this), although his definition of "cardinal number" was almost certainly more precise, even at this early stage of the development of set theory.

Cantor's inconsistency came from the fact that his theory was not built from solid axioms. The way he described sets, it was allowable to think of the set of all sets, and Russell exploited this in his famous paradox (What is the set of all sets that are members of themselves?). There was also a paradox with ordinal numbers, but we don't need to go there. It took some time to get around these paradoxes (there are a couple of ways of doing this), but set theory had to rest on solid axioms first.


Hmmmm. Firstly, why would me not being able to tell you where the set of all primes is (which I have, you just don't like my answer) have anything to do with my understanding of Cantor's diagonal proof?

Because if you do not understand why your counterexample is wrong, then it is not very probable that you understand some of the finer points of set theory.

I cannot explain in clearer terms why your argument is wrong. Perhaps it is my own inability to express myself. However, there is a solution-- send your counterexample to any PhD in mathematics and have them look at it. After all, if it works, you will be quite famous.


First off, there is no one keeping you here. Secondly, you claim to have proven it, and I have stated what some of my objections are.

I have seen no objections other than a ludicrous attempt to ignore the definitions of the terms involved. Once these definitions are in place, there is little room for argument.

I did not like the way you explained Cantor's argument (although it seemed like you understood the basics of it). I would explain it as follows:

The only definition we need for the bulk of the proof is "equivalent." We want to show that N are (0,1) are not equivalent. So, we have one assumption.

Assumption: Assume that N and (0,1) are equivalent.

By the definition of equivalent, our assumption implies that there is a bijection between N and (0,1). We can then construct a real number that is not in the range of the bijection. This proves that the sets are not equivalent, since every element of (0,1) must be in the range of a bijection). At this point, the bulk of the proof is finished. Then by the definition of cardinal number, the cardinal numbers of the two sets are different. Furthermore, the cardinal number of (0,1) is obviously greater than the cardinal number of N (which I can show, if you want me to), so by the definition of [i]countable[i/], (0,1) is not countable.

So, hopefully you see that it is critically important that we agree on the definitions-- if we agreed that the proof was valid and we were simple presenting it, then we could hand-wave some of the definitions, but since you want to get very precise, we must use precise definitions.

[Brian]

Hello Rand!

Thank you for your explanation on how you determined the cardinality of my set. I really do appreciate it, and have a much better understanding of what you mean by cardinality.

With that said, I do not appreciate your misrepresentation of me. Not only is it rude, as you yourself have admitted, but it is dishonest as well. You represented me as asking (or perhaps insisting) that we use “notions that are not definitions.” I have never made such a request. You have characterized it as such (which is a mischaracterization), but the definitions that I have asked you to use are definitions that have valid references. You may not agree with them; they may not be rigorous enough for you, but respectable mathematicians certainly gave the definitions I have asked to use. As I have already said, I will not purposely misrepresent you no matter how much I disagree with you. And if I do inadvertently misrepresent you, when it is brought to my attention I will certainly correct it. My case in point would be my mistaken charge of you not being orthodox in your view of the omnipotence of God. I only ask the same in return. I require some sort of response from you concerning this, or I will walk away from the discussion.

For the sake of discussion (since you will not talk further unless I accept) I will accept your definitions for equivalency and cardinality. However, I would like to note that I do not think they are valid. The reasons being the same reason I do not think SD is valid. With that said, I accept the responsibility of demonstrating there invalidity.


Cantor's almost assuredly understood cardinality to represent the number of elements in a set (the concept of cardinality tells us this), although his definition of "cardinal number" was almost certainly more precise, even at this early stage of the development of set theory.

I do know that Cantor used the definition that 2 sets have the same cardinality iff there exists a bijection between them. However, he also understood this to mean that two sets have the same cardinality iff they are the same size (i.e. they have the same number of elements).


Because if you do not understand why your counterexample is wrong, then it is not very probable that you understand some of the finer points of set theory.

This does not answer my question. My question was, “why would me not being able to tell you where the set of all primes is (which I have, you just don't like my answer) have anything to do with my understanding of Cantor's diagonal proof?” Cantor’s proof is short, elegant, and simple. I’ll bet that there are many high school students who can understand Cantor’s proof. Understanding Cantor’s proof does not require an understanding of the finer points of set theory. I believe I am proof of that. So the question still remains, why would me not being able to tell you "where the set of all primes is" have anything to do with my understanding of Cantor's diagonal proof?


I have seen no objections other than a ludicrous attempt to ignore the definitions of the terms involved. Once these definitions are in place, there is little room for argument.

The argument is that the definitions used are not valid, and have never been shown to be valid. They assume that the set of all sets that are in bijection with each other, and the set of all sets whose cardinality is the same are the same set. This is assumed, and as such has never been proven. I realize you define it that way, but these definitions did not consider the possibility of a set that is not in bijection with another set, and yet they still have the same cardinality. Kerh's paper claims to have demonstrated such a case by using Cantor's Union of Countable Sets Theorem.


I did not like the way you explained Cantor's argument (although it seemed like you understood the basics of it).

First off, you have presented two representations of Cantor’s proof, both of which I agree are fine representations. My proof, which seems to say the exact same thing as your does, was definitely a more descriptive representation of what is exactly happening in the proof. I also included several observations about it. So, if you meant you did not like the aesthetics of my proof that is fine. However, if you thought my proof was mistaken or in error please tell me. Otherwise, I will assume you agreed to everything I said.

Sincerely,

Brian

P.S. As I stated earlier, I probably will not be able to post again until Thursday or Friday.

[Rand]

Today @ 05:18 AM post located here (http://www.theologyweb.com/forum/showthread.php?s=&postid=137268#post137268)
Brian:
For the sake of discussion (since you will not talk further unless I accept) I will accept your definitions for equivalency and cardinality. However, I would like to note that I do not think they are valid. The reasons being the same reason I do not think SD is valid. With that said, I accept the responsibility of demonstrating there invalidity.

I am sorry if I was rude in my previous comments, but now that we have definitions for the terms, we can proceed, and I will show the validity of the definitions.

The definitions that I have given to the terms are all well-defined, and this can be shown if need be:

Definition: Two set are said to be equivalent iff there exists a bijection between them.

This is well defined, because set equivalence has the properties of an equivalence relation (it is not an equivalence relation in actuality, since equivalence relations are defined over sets, and set equivalence is defined over the class of all sets). Set equivalence acts like an equivalence relation in three ways:
It is reflexive, in that any set is equivalent to itself, as seen by the identity mapping.
It is symmetric, since if A is equivalent to B, then B is equivalent to A. This can be seen by looking at the inverse of the bijection that exists between A and B.
It is transitive, since if A is equivalent to B and B is equivalent to C, then A is equivalent to C. This can be seen my looking at the composition of the bijections that map A onto B and B onto C.
Thus, the definition of set equivalence will separate the class of all sets into disjoint classes of mutually equivalent sets.

The next definition that matters is the one for "cardinal number." Here are the two easy definitions:

Definition(1): The cardinal number of a set is an arbitrary representative from the class of sets equivalent to it.

Definition(2): The cardinal number of a set is the class of sets equivalent to it.

From the definition we gave for equivalence, we can see that "cardinal number" is well defined, since each set belongs to a unique class of sets that are mutually equivalent.

Now, at this point, it would be beneficial to show exactly how cardinal numbers are compared. We can show that for any two cardinal numbers, m and n, exactly one of the following holds (trichotomy law):
m=n m>n m<n
It takes a little bit of work to show, but since it is not completely necessary for the topic at hand, we can forego this development (which would involve introducing the ordinal numbers).

Now we come to the definition of "countable," remembering that the cardinal number of the set of natural numbers is Aleph_0.

Definition(1): A set is said to be countable iff its cardinal is less than or equal to Aleph_0.

This definition is well-defined as well, since we have the trichotomy law for the comparison of cardinal numbers. This definition has the advantage of grouping the finite sets with the countably infinite sets, since the concept of countable implies that the finite sets should be considered countable. Another definition, which is also well-defined, is as follows:

Definition(2): A set is said to be countable iff it is finite or its cardinal number is Aleph_0.

Either definition will work, but in this conversation, we have assumed that all of the sets we work with are infinite, so it does not really matter. It is enough to know that an infinite set is countable iff it is equivalent to the set of natural numbers.

With these (well-defined) definitions, we can understand the way "equivalence" and "cardinal number" are related. It seems to me from your last reply that you want to drive a wedge between equivalent sets and sets that have the same cardinal number. Let me know if this is not correct, but it seemed to be the case when you introduced the possibility of a bijection not existing between two sets and yet those sets having the same cardinal number. However, this cannot be done, and I can now show this without a problem (which is the reason precise definitions were needed):

Theorem: Two sets are equivalent iff they have the same cardinal number.
Proof: Assume that sets A and B are equivalent. Then from the definition of "equivalent," A and B are in the same class of mutually equivalent sets. From the definition of Russell and Frege for "cardinal number," this implies that A and B have the same cardinal number. From the first definition given above, we still get the same implication, since the arbitrary sets chosen to represent the cardinal numbers of A and B come from the same class.
Now assume that two sets, A and B, have the same cardinal number. From the definition of "cardinal number", this means that A and B are in the same class of mutually equivalent sets. Thus A and B are equivalent.

Now we can go back to Cantor's proof and examine it more closely. Cantor proved that the set of natural numbers and the interval (0,1) are not equivalent. From what we have just shown, this means that they have different cardinal numbers. The only remaining question is to determine whether or not a<c or a>c, where a and c represent the cardinal numbers of the set of natural numbers and the interval (0,1), respectively. Of these two, it is obvious that a<c (this can be shown, as well). So, since a<c, sets with the cardinal number c are uncountable, from the definition we gave above.

The same process happens with the power set theorem. Since it is shown that A is not equivalent to P(A), we are forced to conclude that the cardinal number of any set is less than the cardinal number of its power set.

[Brian]

Hello Rand!

This will be short and quick, and I hope to post again tomorrow or Friday.


I am sorry if I was rude in my previous comments...

Your apology is accepted. Thank you. I promise to the best of my ability not to misrepresent anything you say. With that said, I understand the definitions and theorems you have put forth. I also agree that they follow from the earlier definitions I agreed to.


It seems to me from your last reply that you want to drive a wedge between equivalent sets and sets that have the same cardinal number.

Yes, you have understood my position. I understand from the definitions you have provided that this possibility is absolutely precluded. However, I think Kerh's paper is correct, and consequently I will try and demonstrate that Cantor's proof fails because it fails to compare his NN to every element in the listing.

Sincerely,

Brian
P.S. I assume you agree with my proof and the comments I made concerning it.

[Rand]

Yesterday @ 09:18 PM post located here (http://www.theologyweb.com/forum/showthread.php?s=&postid=137938#post137938)
Brian:
P.S. I assume you agree with my proof and the comments I made concerning it.

I am not sure what proof you are referring to... If it is to your argument that the set of prime numbers is equivalent to the power set of the prime numbers, then no-- that is not a counterexample. I will try to think of another way to explain why it is wrong, but until then, I cannot comment on it.

If you mean some other proof, then please let me know which one it is.

[Brian]

Hello Rand!


I am not sure what proof you are referring to...

The proof I was referring to was my version of Cantor's proof for the uncountability of R. Included in this proof were some observations concerning the proof. You said you did not like my proof. I have asked you if your objections were asthetic in nature, or if you took issue with something I said. Since I am going to build my case from some of the observations I have made, I would like to find out if you agree with my proof and observations or if you do not. If not, please let me know what parts you take issue with and why. If you agree to what I have said concerning Cantor's proof I will move forward.

Concerning my attempted counter example, I will try and establish the argument once you let me know about my observations and formulation concerning Cantor's proof. By the way, you still have not explained to me why my not being able to tell you where the set of all primes is in my list makes me not able to understand Cantor's proof. By your silence can I assume that you mis-spoke? I have asked questions of you from time to time, and sometimes they go unanswered, ex. my question concerning the nature of your objection to my presentation of Cantor's proof. When I ask a question, can you give me the courtesy of responding? Thanks!

Sincerely,

Brian

[Rand]

I have copied your explanation of Cantor's proof so that I can comment on it.


The DT (diagonal theorem) starts out with the assumption of “given any countable listing of R(0,1).” It then places N in bijection with this set.

No-- the argument starts off with the assumption that there exists a bijection between N and (0,1).


At this point DT asks, “Can every element of R(0,1) be in this countable listing?” To answer this question the DT creates an element of R(0,1), the NN (new number), which is not mapped to by any element of N. It does this with a sequence and algorithm. The algorithm I will use is: “if the nth digit of the nth element of the list is not a ‘5,’ then the nth digit of the NN will be a ‘5,’ otherwise the nth digit of the NN will be a ‘6.’

This part seems OK to me, although a better what to phrase the question would be, "Is every element of (0,1) in the range of the bijection?" We have to remember that the 'listing' is not as important as the bijection itself, because the existence (or lack of) a bijection tells us about set equivalence (and consequently cardinal numbers).


Once this work is done, DT uses DOU. Since no specific element of N taken from the listing maps to the NN, therefore N cannot map onto all of the elements of the listing, and R(0,1) cannot be placed into bijection with N. Therefore, by using DOU it is concluded that R(0,1) is uncountable. Note: Cantor’s proof relies completely on DOU.

"DOU" is not very important. We only have to show that the bijection that we assume existed is not really a bijection after all. This tells us one thing-- N and (0,1) are not equivalent. That is all we have to worry about. Once we are at this point, the proof is over. We can worry about countability later, but once it is proved that the two sets are not equivalent, we have also shown that their cardinal numbers are different.


Concerning my attempted counter example, I will try and establish the argument once you let me know about my observations and formulation concerning Cantor's proof. By the way, you still have not explained to me why my not being able to tell you where the set of all primes is in my list makes me not able to understand Cantor's proof.

If you have indeed constructed a counterexample, then your argument hinges on having a bijection between the set of natural numbers and the power set of the prime numbers. The set of all prime numbers is a subset of the prime numbers, so there must exist some natural number that maps to it. More clearly, using some mathematical notation, we have:
Let g be a bijection from N to P(P).
P is itself a subset of P, so P is an element of P(P).
So there exists an x in N such that g(x)=P.
So I must ask, what x in N maps to P? Obviously there is none, and your arguement is completely worthless. But, since you honestly believe that your argument works, it is hard for me to believe that you understand what is going on.

This is not an indictment against your character or intelligence. It is simply an observation that your mathematical skills must be sub-par if you actually think your argument has merit.

Also, have you noticed that your 'argument,' if it were correct, would work just as well to show that the power set of N is countable? The same method you proposed for the primes works as follows on the set of natural numbers:
{{1},{2},{1,2},{3},{1,3},{2,3},{1,2,3},{4},...}
Of course, this listing fails for the same reason your listing failed-- the set of all natural numbers is not in my list. :)

[Brian]

Hello Rand!


No-- the argument starts off with the assumption that there exists a bijection between N and (0,1).

Since you are nit picking, the argument necessarily starts with the assumption that R is countable. Both what you said and what I said follow from this.


We have to remember that the 'listing' is not as important as the bijection itself, because the existence (or lack of) a bijection tells us about set equivalence (and consequently cardinal numbers).

Please explain to me the nature of this bijection. The diagonal theorem algorithmically "works on" the two dimensional array, which is necessarily the listing. This two dimentional array, where the columns and the rows are necessarily linked together by the diagonal algorithm, is absolutely vital.


"DOU" is not very important.

What? DOU basically says that "if there does not exist a bijection with N, then the infinite set is uncountable." This is equivalent to the second implication of your definition for equivalence, which is "if a bijection does not exist, then the two sets are not equivalent." Cantor relies completely on this part of your definition. Therefore, DOU is vital.


If you have indeed constructed a counterexample, then your argument hinges on having a bijection between the set of natural numbers and the power set of the prime numbers.

If I have indeed constructed a counter example, then the power set theorem is not valid. The essence of the proof would be...

Prove A: The Power Set Theorem is not valid.
Assume ~A: The Power Set Theorem is valid.
~A-->B:|P(P)|>|P|
~B: By counter example |P(P)|=|P|. Contradiction.
~~A by M.T.
A

QED.


But, since you honestly believe that your argument works, it is hard for me to believe that you understand what is going on.

If my counter example is valid, then the power set theorem is not valid. If you have a problem with this, then show me which step in my proof above is incorrect. You still have not shown how "my not being able to tell you where the set of all primes is in my list" has anything to do with my understanding of Cantor's proof. Your claim just does not follow.


This is not an indictment against your character or intelligence. It is simply an observation that your mathematical skills must be sub-par if you actually think your argument has merit.

Compared to you, I am sure that my mathematical skills are "sub-par." However, if my counter example is valid, then my argument has merit. I believe it is you that is making a mistake. This not a reflection of your mathematcial skills, which I am sure are prodigious. Rather, this is a reflection of your humanity. We all make mistakes. :smile:

Sincerely,

Brian

[Rand]

I thought you were going to show where there was a problem with Cantor's proof? Since none of the questions you asked during the last post were very important, I will wait for your critique.

...and just to make sure we are on the same page, here is a version of Cantor's proof (without any reference to uncountable sets or a listing of elements).

Theorem: N is not equivalent to (0,1).
Proof:
Assume N~(0,1)
(1)-> there exists a function, g:N->(0,1), where g is one-to-one and onto.
Define a function in two variables as follows: f:N*R->N defined by the rule, f(n,r)= "the nth post-decimal digit of r."
Define a real number in the continuum, where the nth post-decimal digit is defined to be (f(n,g(n))-1)MOD 10 for each n in N.
By definition, this number defined in (3) is different from every number in the range, and yet it is in (0,1).
By (4), g is not onto.
By (5), N and (0,1) are not equivalent.

Once this result is established, we can ask further questions.

[Brian]

Hello Rand!

I have decided not to continue with this discussion. I do not have the patience, nor do I wish to invest the time. Allow me to explain. Your last post said…


I thought you were going to show where there was a problem with Cantor's proof? Since none of the questions you asked during the last post were very important, I will wait for your critique.

This has happened before. It seems when you are backed into a corner, rather than admit the difficulty you ignore it or choose to no longer talk about it. Let’s start with your criticism of the initial argument that began this thread. You were highly critical, yet when I began to poke holes into your position you told me that you did not want to discuss it further. Then an interesting thing developed. Someone whom you respect puts forth a very similar argument to the one your were critiquing. You immediately bowed to his expertise. I caught the inconsistency and began to press it, and once again you choose not to discuss it. I still do not understand how your buddy’s description of omnipotence is such a “strict view of omnipotence” that you “would have to say that God is not omnipotent.” His description seemed to be completely consistent with orthodoxy. But alas, you don’t want to talk about this inconsistency.

Now I have begun to press another point, which is very much a part of my critique of Cantor’s proof, and you declare that my questions are not “very important,” and therefore you won’t answer them. Beyond the fact that this is just plain discourteous, I ask how would know the importance of my questions if you do not know where my arguments are leading? It has become obvious that you are not interested in having a sincere discussion. Anytime I begin to make a point, especially if it points out your inconsistencies and mistakes, you just ignore it. I have asked you to answer the questions I put forward, but you are unwilling. I would rather spend my online time with people who want to have honest discourse.

Sincerely,

Brian

[Rand]

Today @ 09:23 PM post located here (http://www.theologyweb.com/forum/showthread.php?s=&postid=139570#post139570)
Brian:
I have decided not to continue with this discussion.

Good-- I saw that it was going nowhere, as often happens when people comment on things they know little about.