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mattdamore
02-17-2017, 10:35 AM
I think there's a tendency to be so enamored with a philosophical argument that we tend to not treat certain concepts with as much rigor as they deserve. One concept that has always fascinated me (but which I wished I knew more) is that of the Infinite. The Infinite is most popularly invoked (in my experience) in the context of various cosmological arguments for God's existence. But what I notice is that there are objections as to how the Infinite is being applied in such contexts. So I'd like the purpose of this thread to be an exploration of this very interesting concept: the Infinite.

Before I really get into this, I'd like to construct what is called a set-theoretical hierarchy of numbers, from zero to trans-finite numbers, as inspired by A.W. Moore's The Infinite (1990).

- 0 - [Empty Set] (No sets above here)
- 1 - [Set of one member]
- 2 - [Set with two member]
- 3 - [Set with three members]
- 5 - [There are 65,536 sets above this point]
- 6 - [The number of sets above here has 20,000 digits]
-------------
At this point, I need to use characters for the Greek alphabet, which I don't have. I'll use an English transliteration, but I apologize if this is cause for confusion. I'll try my best to be as clear as I can.

- Omega = Aleph null - [Omega is the first infinite ordinal. Aleph null is first infinite cardinal: the set of all natural numbers. The set of all natural numbers is countably infinite. - The sets above this point are all finite.]

- Omega + 1 - [The rules for addition are different for transfinite numbers. It is actually at this point where Hilbert Hotel becomes an issue. For brevity, I'll stop here.]

- Omega X 2 - [The rules for multiplication are also different for transfinite numbers. Here and above is suppose to be the last point where it's not necessary to use sets.]

- Omega2 - [These are limit ordinals, bigger than ordinals from infinite sets.]

- OmegaOmega - [This is the least ordinal that's bigger than all the natural powers of omega.]

- OmegaOmegaOmega - [It cannot be expressed by an infinite amount of natural powers of omega.]

- Epsilon 0 - [Epsilon null. The first inaccessible ordinal.]

- Aleph 1 - [a. First uncountable ordinal. b. Second infinite cardinal.]

- Aleph Omega - [First cardinal that is preceded by an infinite amount of cardinals.]

- Kappa is the first cardinal such that Kappa = Aleph Kappa

- At this point, the axioms of Zermelo-Fraenkel set theory cannot be used to prove sets beyond this point.

- Here are supposed to be inaccessible cardinals.

As I said, I am a rank amateur when it comes to this concept, and I have probably made mistakes above. Any contribution or clarification is most welcome.

Leonhard
02-17-2017, 10:48 AM
Technically speaking you're not doing any construction at all, you're listing a bunch of results without any technical steps involving in getting from one point to another.

mattdamore
02-17-2017, 11:19 AM
Technically speaking you're not doing any construction at all, you're listing a bunch of results without any technical steps involving in getting from one point to another.

Fair enough. Construction was probably the wrong word. I admitted that much of this is beyond my ken. That's why I started the thread. If anyone could help connect the dots, I'd really appreciate it.

Boxing Pythagoras
02-17-2017, 12:01 PM
At this point, I need to use characters for the Greek alphabet, which I don't have. I'll use an English transliteration, but I apologize if this is cause for confusion. I'll try my best to be as clear as I can.Most fonts support Greek and Hebrew characters, these days, but you'll have to use some extra manner of accessing them-- for example, the Charmap program in Windows or http://typegreek.com/

Another option, and the one which I prefer, is to use LaTeX formatted images. This tool is very helpful in that regard: http://www.codecogs.com/latex/eqneditor.php


Omega = Aleph null - [Omega is the first infinite ordinal. Aleph null is first infinite cardinal: the set of all natural numbers. The set of all natural numbers is countably infinite. - The sets above this point are all finite.]It is not quite true that http://latex.codecogs.com/gif.latex?%5Comega%3D%5Caleph%20_0. The cardinality of omega is Aleph null, but ordinals and cardinals are very different sorts of numbers. We can't just equate them in this way. For example, it is true that http://latex.codecogs.com/gif.latex?%5Caleph%20_0%20%3D%5Caleph%20_0%5E2; however, we know that http://latex.codecogs.com/gif.latex?%5Comega%20%3C%20%5Comega%5E2. Saying that these two numbers equal one another would make our mathematics inconsistent.


As I said, I am a rank amateur when it comes to this concept, and I have probably made mistakes above. Any contribution or clarification is most welcome.In general, it is useful to note the difference between ordinal and cardinal numbers. Ordinals, as their name implies, are a description of how elements of a set may be ordered. Cardinals, on the other hand, are a description of how the elements of one set can be mapped onto another. So, ω describes the first number which is ordinally greater than any Natural number, in transfinite arithmetic. On the other hand, http://latex.codecogs.com/gif.latex?%5Caleph_0 represents the cardinality of any set which can be mapped with 1-to-1 correspondence onto the Natural numbers.

mattdamore
02-17-2017, 01:23 PM
Most fonts support Greek and Hebrew characters, these days, but you'll have to use some extra manner of accessing them-- for example, the Charmap program in Windows or http://typegreek.com/

Another option, and the one which I prefer, is to use LaTeX formatted images. This tool is very helpful in that regard: http://www.codecogs.com/latex/eqneditor.php

Hello Boxing Pythagoras. I really appreciate those links. That will help tremendously.


It is not quite true that http://latex.codecogs.com/gif.latex?%5Comega%3D%5Caleph%20_0. The cardinality of omega is Aleph null, but ordinals and cardinals are very different sorts of numbers. We can't just equate them in this way. For example, it is true that http://latex.codecogs.com/gif.latex?%5Caleph%20_0%20%3D%5Caleph%20_0%5E2; however, we know that http://latex.codecogs.com/gif.latex?%5Comega%20%3C%20%5Comega%5E2. Saying that these two numbers equal one another would make our mathematics inconsistent.


I understand. The way I explained myself was probably misleading. When I used the "=" symbol, I had meant it to mean the "is" in "the cardinality of omega is Aleph null." But I do understand that ordinals and cardinals are different sorts of numbers. Thank you for emphasizing that for me. One question, though, on your example. What is the "2" in http://latex.codecogs.com/gif.latex?%5Caleph%20_0%20%3D%5Caleph%20_0%5E2? Is it raising http://latex.codecogs.com/gif.latex?%5Caleph_0 to the second power? In this case, is it the point that both have the same cardinality? If so, I see your point that because http://latex.codecogs.com/gif.latex?%5Comega%20%3C%20%5Comega%5E2 is the case, cardinality is distinct from ordinality.


In general, it is useful to note the difference between ordinal and cardinal numbers. Ordinals, as their name implies, are a description of how elements of a set may be ordered. Cardinals, on the other hand, are a description of how the elements of one set can be mapped onto another. So, ω describes the first number which is ordinally greater than any Natural number, in transfinite arithmetic. On the other hand, http://latex.codecogs.com/gif.latex?%5Caleph_0 represents the cardinality of any set which can be mapped with 1-to-1 correspondence onto the Natural numbers.

Correct. Clear explanations. Ordinals (the name contains "ordin . . .") relate to order. ω "comes after" (ordinal"ly") the last finite cardinal, and is the first infinite cardinal. And because this cardinal is infinite, it can be put into a one-to-one correspondence with the natural numbers.

P.S. I'm having trouble seeing where the Aleph Null is on the links you provided. Thank you for your insights! You seem to have a lot of knowledge of mathematics.

Boxing Pythagoras
02-17-2017, 01:54 PM
Hello Boxing Pythagoras. I really appreciate those links. That will help tremendously.My pleasure, of course!


One question, though, on your example. What is the "2" in http://latex.codecogs.com/gif.latex?%5Caleph%20_0%20%3D%5Caleph%20_0%5E2? Is it raising http://latex.codecogs.com/gif.latex?%5Caleph_0 to the second power? In this case, is it the point that both have the same cardinality? If so, I see your point that because http://latex.codecogs.com/gif.latex?%5Comega%20%3C%20%5Comega%5E2 is the case, cardinality is distinct from ordinality.Yep, the superscripted 2's imply power operations on their respective numbers. So, the square of http://latex.codecogs.com/gif.latex?%5Caleph_0 is equal to http://latex.codecogs.com/gif.latex?%5Caleph_0, while the square of ω is greater than ω.


Correct. Clear explanations. Ordinals (the name contains "ordin . . .") relate to order. ω "comes after" (ordinal"ly") the last finite cardinal and is the first infinite cardinal.Two small corrections, here. Firstly, we're comparing infinite ordinals to finite ordinals in this case, not cardinals to cardinals-- probably just a minor typographical error, there. However, more importantly, there is no "last" finite ordinal. For any finite ordinal, m, it will always be true that there exists another finite ordinal, n, such that m<n. However, for any finite ordinal, n, it is always true that n<ω.


And because this cardinal is infinite, it can be put into a one-to-one correspondence with the natural numbers.In set theory, the ordinals are defined as sets. The ordinal ω is the set which contains all the Natural numbers as its elements. Therefore, the cardinality of ω is the cardinality of the set of Natural numbers, that is http://latex.codecogs.com/gif.latex?%5Caleph_0.


P.S. I'm having trouble seeing where the Aleph Null is on the links you provided.In LaTeX code, you can denote an Aleph symbol by typing \aleph, and the subscript is denoted by using an underscore. So, for http://latex.codecogs.com/gif.latex?%5Caleph_0, you would type \aleph _0


Thank you for your insights! You seem to have a lot of knowledge of mathematics.I'm still woefully amateurish, but I do have a particular love for mathematics dealing with infinities.

shunyadragon
02-18-2017, 06:33 AM
The problem of the 'concept of the infinite' from the perspective of being 'invoked in the context of various cosmological arguments for God's existence' is that the different 'concepts of infinity' are descriptive, as with all math, of our physical existence, and not definitive as to the limits and nature of our physical existence.

This leads to the problem of apologetic cosmological arguments using the math of 'actual infinities' to define limits of our physical existence. The odd assertion that 'actual infinities' do not exist in the reality of our physical existence, and limit the 'potential infinity of our existence,' is in contradiction with fact that the math of 'actual infinities' is indeed used as part of the science 'tool box,' like all concepts, proofs and axioms, to describe aspects of our physical existence. The nature and application actual infinities sets have no relationship to the question of whether our physical existence is potentially infinite or not,

This is true of axioms of Zermelo-Fraenkel set theory, which was developed to demonstrate a set theory that is free form paradox's such as Russell's Paradox. It is an important set theory, but one of many, and there are many versions and variations developed since. My question is; How do you propose to use Zermelo-Fraenkel set theory to develop your 'concept of infinities?'

Boxing Pythagoras
02-18-2017, 08:11 AM
The odd assertion that 'actual infinities' do not exist in the reality of our physical existence, and limit the 'potential infinity of our existence,' is in contradiction with fact that the math of 'actual infinities' is indeed used as part of the science 'tool box,' like all concepts, proofs and axioms, to describe aspects of our physical existence.I don't think that Matt was attempting to make an assertion one way or the other about the reality of actual infinities. He was simply noting that he was brought to an interest in getting a better understanding of the concept of infinity due to the appearance of that concept in apologetics.


This is true of axioms of Zermelo-Fraenkel set theory, which was developed to demonstrate a set theory that is free form paradox's such as Russell's Paradox. It is an important set theory, but one of many, and there are many versions and variations developed since. My question is; How do you propose to use Zermelo-Fraenkel set theory to develop your 'concept of infinities?'ZFC takes the existence of infinite sets axiomatically. After that, it's just a matter of discovering the properties of such sets.

37818
02-18-2017, 08:47 AM
Infinity from our finite point of view can be never. Parallel lines never meet. Parallel lines meet at infinity. That is just looking at infinity from one aspect.

Boxing Pythagoras
02-18-2017, 09:35 AM
Infinity from our finite point of view can be never. Parallel lines never meet. Parallel lines meet at infinity. That is just looking at infinity from one aspect."Never" is quite a different concept than is "infinity."

To say, "parallel lines never intersect," is not the equivalent of saying, "parallel lines intersect at infinity." These are, in fact, completely opposite statements.

shunyadragon
02-18-2017, 09:38 AM
I don't think that Matt was attempting to make an assertion one way or the other about the reality of actual infinities. He was simply noting that he was brought to an interest in getting a better understanding of the concept of infinity due to the appearance of that concept in apologetics.

OK. No assumption on Matt's purpose, which was unclear.



ZFC takes the existence of infinite sets axiomatically. After that, it's just a matter of discovering the properties of such sets.

True, I was just trying get context of Matt's direction of discussion.

Boxing Pythagoras
02-18-2017, 12:15 PM
Here's a really quick, rather imprecise summary of the idea underlying the infinite ordinals. Let's say that we are attempting to construct a set, but we currently have no elements to place in that set. We can still make meaningful reference to such a set-- and, indeed, we do. We call this the Empty Set, denoted as http://latex.codecogs.com/gif.latex?%5Cemptyset or { }.

Now, we want to create another set. Thankfully, we now have something we can place in this new set-- the Empty Set which we created before. So our new set is {http://latex.codecogs.com/gif.latex?%5Cemptyset}.

If we were to make another set, there are now more elements which we can place in it: the Empty Set, and the set containing the Empty Set, denoted {http://latex.codecogs.com/gif.latex?%5Cemptyset, {http://latex.codecogs.com/gif.latex?%5Cemptyset}}.

We can continue this process over and over. The sets which we create in this manner represent the Natural numbers. The Empty Set represents zero. The set containing zero is one. The set containing zero and one is two. Et cetera, et cetera. There is a very simple way, now, for us to create an ordering on the sets which we have created-- if one set is an element of another set, then the former is "less than" the latter.

Now, let's consider the set which contains all of the Natural numbers, ω. Since any Natural number, n, is an element of ω, it is clear that n<ω. Thus, we have created a number which is greater than any of the Natural numbers.

37818
02-18-2017, 12:24 PM
"Never" is quite a different concept than is "infinity."

To say, "parallel lines never intersect," is not the equivalent of saying, "parallel lines intersect at infinity." These are, in fact, completely opposite statements.

Yeah. It is an issue of language, and what one understands by the words being used. Both statements "never intersecting" and "intersecting at infinity" being understood to be true. Disallowing that is a matter of one's understanding of truth and a different understanding of infinity.

mattdamore
02-19-2017, 11:26 AM
Yep, the superscripted 2's imply power operations on their respective numbers. So, the square of http://latex.codecogs.com/gif.latex?%5Caleph_0 is equal to http://latex.codecogs.com/gif.latex?%5Caleph_0, while the square of ω is greater than ω.

Correct. I believe I follow this. Thank you.


Two small corrections, here. Firstly, we're comparing infinite ordinals to finite ordinals in this case, not cardinals to cardinals-- probably just a minor typographical error, there. However, more importantly, there is no "last" finite ordinal. For any finite ordinal, m, it will always be true that there exists another finite ordinal, n, such that m<n. However, for any finite ordinal, n, it is always true that n<ω.


Yes. I'm understanding the difference between comparing infinite ordinals with finite ordinals. For example, a finite ordinal might be the calendar week according to which Monday is the first day of the week, and so on. Thus, the ordinal number for the calendar week is 7. It seems to me (and please correct me if I'm wrong) that for any finite collection, the ordinal number and the cardinal number are the same. Thus, the calendar week's cardinal number is also 7.

On the other hand, an infinite cardinality and an infinite ordinality is odd to me. If infinite ordinality is defined in terms of sets, then the ordinality of the set of all natural numbers would be the same as its cardinality, correct? This would be true up until we added (+1), multiplied (*2) or raised the power of (X2), the infinite ordinality. The reason this happens is because addition, multiplication, and "squaring", introduce additional ordinality. Let me know if I have that right.



In set theory, the ordinals are defined as sets. The ordinal ω is the set which contains all the Natural numbers as its elements. Therefore, the cardinality of ω is the cardinality of the set of Natural numbers, that is http://latex.codecogs.com/gif.latex?%5Caleph_0.


Yes. I think I said that above. Let me know if I have this part of it understood. Thanks!


In LaTeX code, you can denote an Aleph symbol by typing \aleph, and the subscript is denoted by using an underscore. So, for http://latex.codecogs.com/gif.latex?%5Caleph_0, you would type \aleph _0


I appreciate it. I am not very tech-savvy so I'm still struggling with how to use the links exactly. I don't mean to bother you.


I'm still woefully amateurish, but I do have a particular love for mathematics dealing with infinities.

I'm relatively new to mathematics, but I do find it particularly arresting. It has a sort of petrified, symphonic quality about it that I wasn't aware of in high school.

mattdamore
02-19-2017, 11:57 AM
The problem of the 'concept of the infinite' from the perspective of being 'invoked in the context of various cosmological arguments for God's existence' is that the different 'concepts of infinity' are descriptive, as with all math, of our physical existence, and not definitive as to the limits and nature of our physical existence.

When you say "not definitive as to the limits and nature of our physical existence", what does this mean exactly? What does it mean for a "concept of the infinite" to not be "definitive as to the limits and nature of our physical existence"? Does it mean that the concept of the infinite doesn't tell us exactly what the limits and nature of physical existence are? And if it doesn't tell us exactly, does it tell us something inexact about its limits and nature?

You do say, however, that the concept of the infinite (within the context of cosmological arguments, of course) are "descriptive" of our physical existence. I hadn't heard this before. Why do you think this?


This leads to the problem of apologetic cosmological arguments using the math of 'actual infinities' to define limits of our physical existence. The odd assertion that 'actual infinities' do not exist in the reality of our physical existence, and limit the 'potential infinity of our existence,' is in contradiction with fact that the math of 'actual infinities' is indeed used as part of the science 'tool box,' like all concepts, proofs and axioms, to describe aspects of our physical existence. The nature and application actual infinities sets have no relationship to the question of whether our physical existence is potentially infinite or not,


Hmmm. I'm having trouble following this. Your first proposition seems to imply that Cosmological Arguments commit the mistake of using an Actual Infinite as a Mathematical Concept to "define limits of our physical existence." I'm not sure what this means. I didn't want to get into the Cosmological Arguments per se. I did want to try and do a conceptual analysis of "the infinite", especially as delimited in mathematics. But your proposition does intrigue me. I'm only not sure exactly what it means. I don't know what it means to say that a mathematical concept defines the limits of physical existence, not to mention a mathematical concept involving "the infinite."

I'd rather hold off on your second proposition until we get a good hold on the concept itself. It appears as though Boxing Pythagoras is the knowledgable one in this respect. I will say that I also find it hard to comprehend the notion of an actual infinite "limiting the potential infinity of our existence." I would interact with this, and it seems as though you're communicating something important; I'm not confident, however, that my interaction would be productive because I can't understand your meaning.

And if you wouldn't mind, I'm also not sure what it means to say that "actual infinities" are a "part of the science 'tool box'". What aspect of our physical existence do scientists illuminate with actual infinities?

As for your last proposition, I'm not sure I follow the point. Is your main point that actual infinities are irrelevant to the question as to whether our physical existence is potentially infinite? If it is, that's fine. But for now, I was hoping that we could first perform a conceptual analysis of "the infinite", first.


This is true of axioms of Zermelo-Fraenkel set theory, which was developed to demonstrate a set theory that is free form paradox's such as Russell's Paradox. It is an important set theory, but one of many, and there are many versions and variations developed since. My question is; How do you propose to use Zermelo-Fraenkel set theory to develop your 'concept of infinities?'

Correct me if I'm wrong, but I believe that Zermelo-Fraenkel set theory defines an infinite set S as that kind of set with a proper subset P, according to which S and P have the same cardinality. Further, it is my understanding that Zermelo-Fraenkel set theory only provides a sort of abstract semantics for talking meaningfully about a purely abstract, mathematical universe of sets. It is my understanding that Zermelo-Fraenkel set theory was never meant to stand as a kind of abstract, geographical map meant to coordinate links between the map and the universe. But I could be wrong. I was actually hoping that we could pick Boxing Pythagoras' brain on this, since I think I may be getting this wrong. Thank you for your input!

mattdamore
02-19-2017, 12:07 PM
Excellent summary of infinite ordinals! Quick question, though.


Now, let's consider the set which contains all of the Natural numbers, ω. Since any Natural number, n, is an element of ω, it is clear that n<ω. Thus, we have created a number which is greater than any of the Natural numbers.

I might have misunderstood your meaning, then, in my other reply to you.

It was my understanding that the set of all natural numbers had a cardinality which was equal to its ordinality. I had thought that "n" becomes "less than" ω only when it's the case that ω+1, or ω*2, or ω2, or ωω? I do remember you saying that cardinals and ordinals are different. Is it that an ordinal and a cardinal could have the same number, but that it's a different kind of number, due to the fact that ordinals and cardinals are different kinds of numbers? Thank you for your patience.

Boxing Pythagoras
02-19-2017, 12:27 PM
Yes. I'm understanding the difference between comparing infinite ordinals with finite ordinals. For example, a finite ordinal might be the calendar week according to which Monday is the first day of the week, and so on. Thus, the ordinal number for the calendar week is 7. It seems to me (and please correct me if I'm wrong) that for any finite collection, the ordinal number and the cardinal number are the same. Thus, the calendar week's cardinal number is also 7.Did you get a chance to read my really quick set theory primer from post #12? It wouldn't quite be accurate to say that the ordinal number for the week is 7, so much as we would say that Saturday is the element of the days of the week which corresponds to the ordinal 7. The set of days of the week can be put in one-to-one correspondence with the elements of the ordinal number 7, so the week has the cardinality of 7.


On the other hand, an infinite cardinality and infinite ordinality is odd to me. If infinite ordinality is defined in terms of sets, then ordinality of the set of all natural numbers would be the same as its cardinality, correct?Nope. Ordinals and Cardinals are not the same sorts of things, despite the fact that we tend to use the same symbols for both when dealing with finite numbers. Again, ordinality tells us about an ordering relationship-- where should this element show up in a list of elements? Cardinality tells us how the elements of one set can be mapped onto another-- can this list be lined up with some other list? So, while ω has cardinality http://latex.codecogs.com/gif.latex?%5Caleph%20_0, nonetheless we cannot say that http://latex.codecogs.com/gif.latex?%5Comega%3D%5Caleph%20_0. So, ω<ω+1, despite the fact that ω and ω+1 have the same cardinality.


I appreciate it. I am not very tech-savvy so I'm still struggling with how to use the links exactly. I don't mean to bother you.Not a bother, at all. Feel free to Private Message me if you have any questions on how I utilize those links. I'll be happy to help.


I'm relatively new to mathematics, but I do find it particularly arresting. It has a sort of petrified, symphonic quality about it that I wasn't aware of in high school.Yeah, the way in which mathematics is often taught in schools rather obfuscates its beauty. Honestly, the best way to come to that sense, in my opinion, is to start looking into the History of Mathematics. Seeing the reasoning and work that went into the inventions and discoveries of mathematics, and the people responsible for these, can really make you appreciate the field.

shunyadragon
02-19-2017, 05:31 PM
When you say "not definitive as to the limits and nature of our physical existence", what does this mean exactly? What does it mean for a "concept of the infinite" to not be "definitive as to the limits and nature of our physical existence"?

[quote] Does it mean that the concept of the infinite doesn't tell us exactly what the limits and nature of physical existence are?

Yes, math over the millennia has been developed as the tool box with science to understand our physical existence and not define it. This has evolved from counting sheep and goats, to the modern math that science uses to describe the sciences, Quantum Mechanics and Cosmology.


And if it doesn't tell us exactly, does it tell us something inexact about its limits and nature? [quote]

No, not its limits, actual infinities and potential infinities, are descriptive of what could be the nature of infinities. They are used in math to describe various aspects of the nature of physical existence within math proofs, theorems, and described in axioms. Infinities were never meant to prove nor demonstrate the limits or our physical existence.

There is no proof that our universe is finite nor infinite, and some way limited nor eternal. It can be assumed by the evidence that it is potentially eternal.

[quote]
You do say, however, that the concept of the infinite (within the context of cosmological arguments, of course) are "descriptive" of our physical existence. I hadn't heard this before. Why do you think this?

No, within math the 'concept of infinity' is descriptive as part of the tool box of science. Within the context of apologetics cosmological arguments the concepts it cannot be used in the logical arguments fro the finite nature of our physical existence.


Hmmm. I'm having trouble following this. Your first proposition seems to imply that Cosmological Arguments commit the mistake of using an Actual Infinite as a Mathematical Concept to "define limits of our physical existence." I'm not sure what this means. I didn't want to get into the Cosmological Arguments per se. I did want to try and do a conceptual analysis of "the infinite", especially as delimited in mathematics. But your proposition does intrigue me. I'm only not sure exactly what it means. I don't know what it means to say that a mathematical concept defines the limits of physical existence, not to mention a mathematical concept involving "the infinite."



You mentioned the issue of infinities in apologetic cosmological arguments. Cosmological arguments use infinities to logically to conclude, define or prove that our physical existence is finite.

I do not believe the conceptual understand of infinities necessarily exists within the of math construction of infinities in axioms, sets and theorems. I understand the discussion between Pythagoras and you, and I bow to Pythagoras as to his deep understanding of the mechanics of math and I will continue to follow the discussion, but I believe the understanding of math and the 'concepts of infinity' go deeper than a course on the nature of set theories and the mechanics of math.



I'd rather hold off on your second proposition until we get a good hold on the concept itself. It appears as though Boxing Pythagoras is the knowledgable one in this respect. I will say that I also find it hard to comprehend the notion of an actual infinite "limiting the potential infinity of our existence." I would interact with this, and it seems as though you're communicating something important; I'm not confident, however, that my interaction would be productive because I can't understand your meaning.

Fine . . . hold off if you wish. I defer to Pythagoras as being more knowledgeable concerning the meahanics of math.



And if you wouldn't mind, I'm also not sure what it means to say that "actual infinities" are a "part of the science 'tool box'". What aspect of our physical existence do scientists illuminate with actual infinities?

Understanding the nature of math being a part of the 'tool box' of science, and more specifically 'actual infinities' used in physics and cosmology I refer to the following, which I may cite from in later posts.

http://www.iep.utm.edu/infinite/



As for your last proposition, I'm not sure I follow the point. Is your main point that actual infinities are irrelevant to the question as to whether our physical existence is potentially infinite? If it is, that's fine. But for now, I was hoping that we could first perform a conceptual analysis of "the infinite", first.

Yes, actual infinities are defined as sets withing a greater reality such as a physical existence that is either infinite or finite, or in the case that it is potentially eternal.

Simply . . .


Actual infinity is the idea that numbers, or some other type of mathematical object, can form an actual, completed totality; namely, a set. Hence, in the philosophy of mathematics, the abstraction of actual infinity involves the acceptance of infinite entities, such as the set of all natural numbers or an infinite sequence of rational numbers, as given objects. This is contrasted with potential infinity, in which a non-terminating process (such as "add 1 to the previous number") produces an unending "infinite" sequence of results, but each individual result is finite and is achieved in a finite number of steps.



Correct me if I'm wrong, but I believe that Zermelo-Fraenkel set theory defines an infinite set S as that kind of set with a proper subset P, according to which S and P have the same cardinality. Further, it is my understanding that Zermelo-Fraenkel set theory only provides a sort of abstract semantics for talking meaningfully about a purely abstract, mathematical universe of sets. It is my understanding that Zermelo-Fraenkel set theory was never meant to stand as a kind of abstract, geographical map meant to coordinate links between the map and the universe. But I could be wrong. I was actually hoping that we could pick Boxing Pythagoras' brain on this, since I think I may be getting this wrong. Thank you for your input!

I will bow to Pythagoras on this and his knowledge of math for the best explanation.

Boxing Pythagoras
02-20-2017, 02:31 AM
And if you wouldn't mind, I'm also not sure what it means to say that "actual infinities" are a "part of the science 'tool box'". What aspect of our physical existence do scientists illuminate with actual infinities?Calculus is a tool for calculating actually infinite sets of objects, and this tool can be used to describe the workings of the real world with incredible accuracy, as Newton demonstrated with his celestial mechanics. There is some philosophical debate, however, as to whether such tools are just decent idealizations which provide reasonable approximations of reality, or whether they accurately describe reality.


Correct me if I'm wrong, but I believe that Zermelo-Fraenkel set theory defines an infinite set S as that kind of set with a proper subset P, according to which S and P have the same cardinality. Further, it is my understanding that Zermelo-Fraenkel set theory only provides a sort of abstract semantics for talking meaningfully about a purely abstract, mathematical universe of sets. It is my understanding that Zermelo-Fraenkel set theory was never meant to stand as a kind of abstract, geographical map meant to coordinate links between the map and the universe. But I could be wrong. I was actually hoping that we could pick Boxing Pythagoras' brain on this, since I think I may be getting this wrong. Thank you for your input!I'm a Formalist when it comes to the philosophy of mathematics, so I think ALL mathematics is a purely abstract means of describing the real world-- including basic arithmetic. However, there are ways in which this abstraction can be more or less accurate in its description of the world.

For ZFC, the easiest example would be whether or not space-time is continuous. If space and/or time is continuous, then ZFC provides a method for discussing complete sets of locations or moments despite their infinitude. If space and time are discrete, then ZFC gives us a far larger toolbox than is necessary to describe these things.

shunyadragon
02-20-2017, 01:30 PM
Calculus is a tool for calculating actually infinite sets of objects, and this tool can be used to describe the workings of the real world with incredible accuracy, as Newton demonstrated with his celestial mechanics. There is some philosophical debate, however, as to whether such tools are just decent idealizations which provide reasonable approximations of reality, or whether they accurately describe reality.

I'm a Formalist when it comes to the philosophy of mathematics, so I think ALL mathematics is a purely abstract means of describing the real world-- including basic arithmetic. However, there are ways in which this abstraction can be more or less accurate in its description of the world.



We share the same view.

mattdamore
02-27-2017, 02:46 PM
Did you get a chance to read my really quick set theory primer from post #12? It wouldn't quite be accurate to say that the ordinal number for the week is 7, so much as we would say that Saturday is the element of the days of the week which corresponds to the ordinal 7. The set of days of the week can be put in one-to-one correspondence with the elements of the ordinal number 7, so the week has the cardinality of 7.

Got it.


Nope. Ordinals and Cardinals are not the same sorts of things, despite the fact that we tend to use the same symbols for both when dealing with finite numbers. Again, ordinality tells us about an ordering relationship-- where should this element show up in a list of elements? Cardinality tells us how the elements of one set can be mapped onto another-- can this list be lined up with some other list? So, while ω has cardinality http://latex.codecogs.com/gif.latex?%5Caleph%20_0, nonetheless we cannot say that http://latex.codecogs.com/gif.latex?%5Comega%3D%5Caleph%20_0. So, ω<ω+1, despite the fact that ω and ω+1 have the same cardinality.


Got it. Cardinality and Ordinality are different kinds of things.


Not a bother, at all. Feel free to Private Message me if you have any questions on how I utilize those links. I'll be happy to help.


Thanks!


Yeah, the way in which mathematics is often taught in schools rather obfuscates its beauty. Honestly, the best way to come to that sense, in my opinion, is to start looking into the History of Mathematics. Seeing the reasoning and work that went into the inventions and discoveries of mathematics, and the people responsible for these, can really make you appreciate the field.

Agreed.

Do you think we have a good enough grasp on the infinite as of right now (even though I realize we're scratching the surface of the surface . . .)?

mattdamore
02-27-2017, 02:47 PM
Excellent summary of infinite ordinals! Quick question, though.



I might have misunderstood your meaning, then, in my other reply to you.

It was my understanding that the set of all natural numbers had a cardinality which was equal to its ordinality. I had thought that "n" becomes "less than" ω only when it's the case that ω+1, or ω*2, or ω2, or ωω? I do remember you saying that cardinals and ordinals are different. Is it that an ordinal and a cardinal could have the same number, but that it's a different kind of number, due to the fact that ordinals and cardinals are different kinds of numbers? Thank you for your patience.

I think I was hinting at your worry here: in terms of ordinals and cardinals being different kinds of numbers. I should have said 'different kinds of things'.

mattdamore
02-27-2017, 02:48 PM
When you say "not definitive as to the limits and nature of our physical existence", what does this mean exactly? What does it mean for a "concept of the infinite" to not be "definitive as to the limits and nature of our physical existence"?



Yes, math over the millennia has been developed as the tool box with science to understand our physical existence and not define it. This has evolved from counting sheep and goats, to the modern math that science uses to describe the sciences, Quantum Mechanics and Cosmology.

And if it doesn't tell us exactly, does it tell us something inexact about its limits and nature?

No, not its limits, actual infinities and potential infinities, are descriptive of what could be the nature of infinities. They are used in math to describe various aspects of the nature of physical existence within math proofs, theorems, and described in axioms. Infinities were never meant to prove nor demonstrate the limits or our physical existence.

There is no proof that our universe is finite nor infinite, and some way limited nor eternal. It can be assumed by the evidence that it is potentially eternal.



No, within math the 'concept of infinity' is descriptive as part of the tool box of science. Within the context of apologetics cosmological arguments the concepts it cannot be used in the logical arguments fro the finite nature of our physical existence.





I do not believe the conceptual understand of infinities necessarily exists within the of math construction of infinities in axioms, sets and theorems. I understand the discussion between Pythagoras and you, and I bow to Pythagoras as to his deep understanding of the mechanics of math and I will continue to follow the discussion, but I believe the understanding of math and the 'concepts of infinity' go deeper than a course on the nature of set theories and the mechanics of math.



Fine . . . hold off if you wish. I defer to Pythagoras as being more knowledgeable concerning the meahanics of math.



Understanding the nature of math being a part of the 'tool box' of science, and more specifically 'actual infinities' used in physics and cosmology I refer to the following, which I may cite from in later posts.

http://www.iep.utm.edu/infinite/



Yes, actual infinities are defined as sets withing a greater reality such as a physical existence that is either infinite or finite, or in the case that it is potentially eternal.

Simply . . .


Actual infinity is the idea that numbers, or some other type of mathematical object, can form an actual, completed totality; namely, a set. Hence, in the philosophy of mathematics, the abstraction of actual infinity involves the acceptance of infinite entities, such as the set of all natural numbers or an infinite sequence of rational numbers, as given objects. This is contrasted with potential infinity, in which a non-terminating process (such as "add 1 to the previous number") produces an unending "infinite" sequence of results, but each individual result is finite and is achieved in a finite number of steps.



I will bow to Pythagoras on this and his knowledge of math for the best explanation.

Thanks for the clarification, Shunyadragon.

Boxing Pythagoras
02-27-2017, 08:44 PM
Do you think we have a good enough grasp on the infinite as of right now (even though I realize we're scratching the surface of the surface . . .)?It depends on what we want to use it to discuss. If we're going to talk about the question of an infinite regress posed by apologists and theologians, then I think we're actually better served by discussing a different concept of infinity than that of Cantor's set theory. The transfinite numbers are wonderfully interesting, to be sure, but transfinite arithmetic is different than common arithmetic, and this can quite easily lead into equivocation and confusion.

To that end, allow me to briefly introduce the Hyperreal numbers.

First, some history. The Natural numbers are so named because that number system was the first to be developed and explored, amongst modern number systems. It involves the whole numbers with which one counts-- 0, 1, 2, 3, 4, et cetera. However, the Natural numbers are incomplete. Think, for example, of the equation http://latex.codecogs.com/gif.latex?5&plus;x%3D4. This fairly simple algebra problem has no solution on the Natural numbers, but we can extend the concept of the Natural numbers to give us numbers which are less than zero-- that is to say, numbers which are negative. This extension gives us the Integers. The Integers, themselves, can be extended in order to solve equations like http://latex.codecogs.com/gif.latex?5%5Ccdot%20x%3D4. For this, we need the Rational numbers. But the Rational numbers are insufficient to solve something like http://latex.codecogs.com/gif.latex?x%5E2%3D5, so we can extend once again to create the Real numbers.

Just as the Reals extend the Rationals, and the Rationals extend the Integers, and the Integers extend the Naturals; so too can we create an extension for the Real numbers. This extension will allow us to discuss infinite numbers with consistency, using the exact same arithmetic and algebra with which we are already familiar. We call this extension the Hyperreal numbers.

On the Hyperreal number system, there exist numbers which are described as being infinite. A Hyperreal number, K, is called infinite if and only if, for all n such that n is a Natural number, it is true that http://latex.codecogs.com/gif.latex?%7Cn%7C%3C%7CK%7C. Any number which is not infinite is called finite. There are also numbers which are called infinitesimal. A Hyperreal number, ε, is infinitesimal if and only if, for all r such that r is a Real number, it is true that http://latex.codecogs.com/gif.latex?%7C%5Cepsilon%7C%3C%7Cr%7C. Note that this means zero is the only Real number which is infinitesimal.

The Hyperreal numbers have some very familiar properties. For example, given an infinite Hyperreal number, K, it will always be true that http://latex.codecogs.com/gif.latex?K-1%3CK%3CK&plus;1. Note that K-1, K, and K+1 are all infinite numbers, but they are different infinite numbers. In exactly the same way, if K is positive, it will also be true that http://latex.codecogs.com/gif.latex?%5Cfrac%7BK%7D%7B2%7D%3CK%3C2K. For a more detailed breakdown of the Hyperreals, check out this link (http://mathforum.org/dr.math/faq/analysis_hyperreals.html), or if you really want to get in depth with the subject, pick up Abraham Robinson's book Non-Standard Analysis. There is no better text on the subject-- after all, it was Robinson who discovered, invented, and developed this number system.

The Hyperreals can be added, subtracted, multiplied, divided, exponentiated, logarithmed, and manipulated in exactly the same ways that the Real numbers can be. They give us a very familiar platform with which to discuss infinities, and as such, they are far more intuitive for use in discussions about infinite regress.

mattdamore
03-07-2017, 06:58 PM
It depends on what we want to use it to discuss. If we're going to talk about the question of an infinite regress posed by apologists and theologians, then I think we're actually better served by discussing a different concept of infinity than that of Cantor's set theory. The transfinite numbers are wonderfully interesting, to be sure, but transfinite arithmetic is different than common arithmetic, and this can quite easily lead into equivocation and confusion.

To that end, allow me to briefly introduce the Hyperreal numbers.

First, some history. The Natural numbers are so named because that number system was the first to be developed and explored, amongst modern number systems. It involves the whole numbers with which one counts-- 0, 1, 2, 3, 4, et cetera. However, the Natural numbers are incomplete. Think, for example, of the equation http://latex.codecogs.com/gif.latex?5&plus;x%3D4. This fairly simple algebra problem has no solution on the Natural numbers, but we can extend the concept of the Natural numbers to give us numbers which are less than zero-- that is to say, numbers which are negative. This extension gives us the Integers. The Integers, themselves, can be extended in order to solve equations like http://latex.codecogs.com/gif.latex?5%5Ccdot%20x%3D4. For this, we need the Rational numbers. But the Rational numbers are insufficient to solve something like http://latex.codecogs.com/gif.latex?x%5E2%3D5, so we can extend once again to create the Real numbers.

Just as the Reals extend the Rationals, and the Rationals extend the Integers, and the Integers extend the Naturals; so too can we create an extension for the Real numbers. This extension will allow us to discuss infinite numbers with consistency, using the exact same arithmetic and algebra with which we are already familiar. We call this extension the Hyperreal numbers.

On the Hyperreal number system, there exist numbers which are described as being infinite. A Hyperreal number, K, is called infinite if and only if, for all n such that n is a Natural number, it is true that http://latex.codecogs.com/gif.latex?%7Cn%7C%3C%7CK%7C. Any number which is not infinite is called finite. There are also numbers which are called infinitesimal. A Hyperreal number, ε, is infinitesimal if and only if, for all r such that r is a Real number, it is true that http://latex.codecogs.com/gif.latex?%7C%5Cepsilon%7C%3C%7Cr%7C. Note that this means zero is the only Real number which is infinitesimal.

The Hyperreal numbers have some very familiar properties. For example, given an infinite Hyperreal number, K, it will always be true that http://latex.codecogs.com/gif.latex?K-1%3CK%3CK&plus;1. Note that K-1, K, and K+1 are all infinite numbers, but they are different infinite numbers. In exactly the same way, if K is positive, it will also be true that http://latex.codecogs.com/gif.latex?%5Cfrac%7BK%7D%7B2%7D%3CK%3C2K. For a more detailed breakdown of the Hyperreals, check out this link (http://mathforum.org/dr.math/faq/analysis_hyperreals.html), or if you really want to get in depth with the subject, pick up Abraham Robinson's book Non-Standard Analysis. There is no better text on the subject-- after all, it was Robinson who discovered, invented, and developed this number system.

The Hyperreals can be added, subtracted, multiplied, divided, exponentiated, logarithmed, and manipulated in exactly the same ways that the Real numbers can be. They give us a very familiar platform with which to discuss infinities, and as such, they are far more intuitive for use in discussions about infinite regress.

I follow you, and I very much appreciate the explanation. Sure! Is it your experience that apologists and theologians neglect to talk about infinite regresses in terms of Hyperreal numbers? Is this a shortcoming in the apologetic? If so, I would really like to dive in to where the apologists are in error with this concept.

Boxing Pythagoras
03-08-2017, 02:42 AM
Is it your experience that apologists and theologians neglect to talk about infinite regresses in terms of Hyperreal numbers?It's not so much that they neglect to talk about it as that they are ignorant of the Hyperreals and how these can resolve many of the typically cited questions regarding infinite regress.

For example, I've written two small articles critiquing William Lane Craig's understanding of infinity. Dr. Craig claims that things like Hilbert's Grand Hotel and Al Ghazali's infinite regress of celestial motion reveal absurdities in the concept of the infinite. I show that the proposed absurdities are actually the result of his misconceptions, and that we can easily resolve the issues which he questions by using the Hyperreals.

https://boxingpythagoras.com/2015/10/16/wlc-doesnt-understand-infinity/
And
https://boxingpythagoras.com/2015/10/23/wlc-doesnt-understand-infinity-part-2/

shunyadragon
03-08-2017, 04:09 AM
It's not so much that they neglect to talk about it as that they are ignorant of the Hyperreals and how these can resolve many of the typically cited questions regarding infinite regress.

For example, I've written two small articles critiquing William Lane Craig's understanding of infinity. Dr. Craig claims that things like Hilbert's Grand Hotel and Al Ghazali's infinite regress of celestial motion reveal absurdities in the concept of the infinite. I show that the proposed absurdities are actually the result of his misconceptions, and that we can easily resolve the issues which he questions by using the Hyperreals.

https://boxingpythagoras.com/2015/10/16/wlc-doesnt-understand-infinity/
And
https://boxingpythagoras.com/2015/10/23/wlc-doesnt-understand-infinity-part-2/

Your articles are excellent! I actually do not consider WLC's bad math simply naive misconceptions. I consider it is a dishonest deceptive misuse of math.

Boxing Pythagoras
03-08-2017, 07:39 AM
Your articles are excellent! I actually do not consider WLC's bad math simply naive misconceptions. I consider it is a dishonest deceptive misuse of math.I try to read opposing arguments with a charitable view whenever there is not an overt reason to believe that they actually understand that their views are not generally accepted but present them as the concrete truth, anyway.

In this case, it seems that Craig is simply ignorant of the actual mathematics and philosophy thereof. He doesn't seem to even understand the mathematics which he does discuss, let alone the vast body of infinite mathematics which he does not discuss.

If, on the other hand, he actually understood the math and demonstrated familiarity with the methods by which that math resolves his proposed absurdities, but proceeded to pretend that the notion of the infinite was still unreconcilable or inconsistent, then I might be inclined to think him dishonest. Since this is not the case, the charitable thing is to simply think he is ignorant of the subject matter.

mattdamore
03-08-2017, 09:42 AM
It's not so much that they neglect to talk about it as that they are ignorant of the Hyperreals and how these can resolve many of the typically cited questions regarding infinite regress.

For example, I've written two small articles critiquing William Lane Craig's understanding of infinity. Dr. Craig claims that things like Hilbert's Grand Hotel and Al Ghazali's infinite regress of celestial motion reveal absurdities in the concept of the infinite. I show that the proposed absurdities are actually the result of his misconceptions, and that we can easily resolve the issues which he questions by using the Hyperreals.

https://boxingpythagoras.com/2015/10/16/wlc-doesnt-understand-infinity/
And
https://boxingpythagoras.com/2015/10/23/wlc-doesnt-understand-infinity-part-2/

I appreciate your links. I'll give those articles a look and have something probably by the weekend. Dr. Craig and I have had a professor/student relationship for the past year, and we've dined together as a class. In one-on-one conversation, he's a really nice, down-to-earth, pleasant guy. Perhaps I could bring these things up and see what he thinks. He's currently working on a conceptual analysis of the Christian Atonement, but he's always willing to talk about these things. I'll have to wait until next Fall until he returns to campus, though!

My problem is that I haven't had much exposure to the way "infinity" is used in that particular project Dr. Craig had. So I have a lot of reading to do. But thank you for perhaps unearthing such misconceptions! I trust your allegation of "misconception" comes from a good, productive spirit, as evidenced by the tone of your posts.

mattdamore
03-08-2017, 09:48 AM
Your articles are excellent! I actually do not consider WLC's bad math simply naive misconceptions. I consider it is a dishonest deceptive misuse of math.

Hmmm. I've known Dr. Craig for the past two semesters at Houston Baptist University, and have gotten to know the wonderful faculty and friends in that particular academic circle, and I can assure you that there's not the least hint - that I've discerned - of deception in any area of academic research and study. I realize I could be wrong, but I'm typically a good reader of character, and after many conversations with Dr. Craig, I can tell you the guy is a very nice, sincere person, who has been very caring toward me, and everyone in our class. I'm sorry you've gotten that impression, however. I really am. It's just hard to hear that said about someone I've known for a year who clearly evidences indicators of a stalwart character. But I don't want to sidetrack the issue from the concept of the "infinite" too much. I just wanted to tell you my perspective, and perhaps we can just agree to disagree.

Boxing Pythagoras
03-08-2017, 10:33 AM
I appreciate your links. I'll give those articles a look and have something probably by the weekend. Dr. Craig and I have had a professor/student relationship for the past year, and we've dined together as a class. In one-on-one conversation, he's a really nice, down-to-earth, pleasant guy.I definitely get that feeling from seeing his interactions with his students, and I've always thought him to be utterly sincere in his discussions. Despite the fact that I disagree with him on a great many things, I do have a great deal of respect for Dr. Craig.


Perhaps I could bring these things up and see what he thinks.That'd be awesome! Thanks!


My problem is that I haven't had much exposure to the way "infinity" is used in that particular project Dr. Craig had. So I have a lot of reading to do.By all means, start by listening to the audio from Dr. Craig's Excursus on Natural Theology , parts 9 and 10 (or watch the video, which I think is available as well). Then read over my articles again to see if I have treated Dr. Craig's arguments fairly. After that, I can help point you to good references which discuss these concepts independently of theology, so you can judge whether I treat them accurately.

mattdamore
03-08-2017, 11:21 AM
I definitely get that feeling from seeing his interactions with his students, and I've always thought him to be utterly sincere in his discussions. Despite the fact that I disagree with him on a great many things, I do have a great deal of respect for Dr. Craig.

That'd be awesome! Thanks!

By all means, start by listening to the audio from Dr. Craig's Excursus on Natural Theology , parts 9 and 10 (or watch the video, which I think is available as well). Then read over my articles again to see if I have treated Dr. Craig's arguments fairly. After that, I can help point you to good references which discuss these concepts independently of theology, so you can judge whether I treat them accurately.

I've already put on the wish-list: The Blackwell Companion to Natural Theology, The Kalam Cosmological Argument, and I think I'll read his articles on his website as well. I'll start with the Excursus on Natural Theology, and get to your articles after that. I have a full load doing Graduate Studies and now I have more homework. No worries, though! I'll fit this in as soon as I can. I'll see this as prep for when I talk to him next about you. I saw that Dr. Craig reviews Jordan Howard Sobel's Logic and Theism: Arguments for and against Belief in God, and Sobel mentions hyperreal numbers. It's strange Dr. Craig wouldn't mention it in the review. But I think I will read through Dr. Craig's review (http://www.reasonablefaith.org/sobels-acid-bath-for-theism-review-article-logic-and-theism) and another article (http://www.reasonablefaith.org/j-howard-sobel-on-the-kalam-cosmological-argument) he wrote on just that portion of the book focused on The Kalam.

Boxing Pythagoras
03-08-2017, 12:32 PM
I'll start with the Excursus on Natural Theology, and get to your articles after that.For the sake of ease, here are the videos and transcripts for Dr. Craig's Excursus parts 9 and 10:

https://www.youtube.com/watch?v=MF_r_zdDFro
Transcript of Excursus Part 9 (http://www.reasonablefaith.org/defenders-3-podcast/transcript/excursus-on-natural-theology-part-9)


https://www.youtube.com/watch?v=dJ-_Ig5YF-4
Transcript of Excursus Part 10 (http://www.reasonablefaith.org/defenders-3-podcast/transcript/excursus-on-natural-theology-part-10)


I have a full load doing Graduate Studies and now I have more homework. No worries, though! I'll fit this in as soon as I can.I fully understand that. I certainly know how difficult it can be to find time for such things. The discussion will definitely be here whenever you're ready for it, though.


I saw that Dr. Craig reviews Jordan Howard Sobel's Logic and Theism: Arguments for and against Belief in God, and Sobel mentions hyperreal numbers. It's strange Dr. Craig wouldn't mention it in the review. But I think I will read through Dr. Craig's review (http://www.reasonablefaith.org/sobels-acid-bath-for-theism-review-article-logic-and-theism) and another article (http://www.reasonablefaith.org/j-howard-sobel-on-the-kalam-cosmological-argument) he wrote on just that portion of the book focused on The Kalam.Very interesting. I've just added Sobel's work to my ever-growing reading list.

shunyadragon
03-09-2017, 02:53 PM
Hmmm. I've known Dr. Craig for the past two semesters at Houston Baptist University, and have gotten to know the wonderful faculty and friends in that particular academic circle, and I can assure you that there's not the least hint - that I've discerned - of deception in any area of academic research and study. I realize I could be wrong, but I'm typically a good reader of character, and after many conversations with Dr. Craig, I can tell you the guy is a very nice, sincere person, who has been very caring toward me, and everyone in our class. I'm sorry you've gotten that impression, however. I really am. It's just hard to hear that said about someone I've known for a year who clearly evidences indicators of a stalwart character. But I don't want to sidetrack the issue from the concept of the "infinite" too much. I just wanted to tell you my perspective, and perhaps we can just agree to disagree.

I do not want to dwell on this to derail the thread which is yours, but it would interesting if you would ask others at your college and WLC to get feed back on why the math of infinity they use in their arguments does not address the concept of hyperreal numbers and contemporary set theory as Boxing Pythagoras describes.

I can understand someone like 'hansgeorge,' a Twebber, who simply upfront denies contemporary math and everything since and including Cantor's set theory, but I do not have an explanation from WLC.

I have a good math background up to the graduate level, but nothing close to Boxing Pythagoras that is why I differ to him here and learn from him concerning math. Based on my background I easily understand the concepts as described by Boxing Pythagoras, and the problems with WLC's math.

You could simply refer to Boxing Pythagoras's essays and ask for an explanation.

Boxing Pythagoras
03-10-2017, 05:44 AM
I do not want to dwell on this to derail the thread which is yours, but it would interesting if you would ask others at your college and WLC to get feed back on why the math of infinity they use in their arguments does not address the concept of hyperreal numbers and contemporary set theory as Boxing Pythagoras describes.To be fair, Set Theory is a particularly confusing area of mathematics, which is why Cantor was so controversial at first. I honestly don't expect non-mathematicians to be fully familiar with it.

The Hyperreal numbers are, unfortunately, a bit more obscure. While I would love to see non-standard analysis replace the ε,δ-limit foundation upon which modern Calculus is taught, it has not yet gained anywhere near as widespread recognition. As such, it is entirely unsurprising that a non-mathematician might be wholly ignorant of the entire field of non-standard analysis.

My problem isn't so much that Dr. Craig is ignorant of these concepts-- most people are. Rather, it is that he has attempted to pronounce on the topic of infinity, and has spread those pronouncements to his rather sizable audience, while making mistakes about very basic properties of the math which he is trying to invoke in order to prove his point. His audience is generally even more ignorant of the math than he is-- which, again, is not meant as any sort of indictment upon them-- so they tend to trust Dr. Craig and his multiple PhD's when he speaks on the subject.


You could simply refer to Boxing Pythagoras's essays and ask for an explanation.As I suggested to Matt, I would like for him to start by seeing what Dr. Craig has to say, then read my work to be sure that I am treating Dr. Craig's argument irenically, and then to other mathematical works to be sure that I am representing the math properly. I don't pretend to be any sort of real authority on the subject. I don't have any fancy letters after my name. I'm just a dude who really, really enjoys mathematics and has a particular fondness for the mathematics of infinity.

By all means, I am perfectly willing to discuss any questions about the work which I have written. However, I wouldn't want anyone to simply ignore those who disagree with me and look to my writings as if they are some absolute truth of the matter. If Matt can bring my complaints to the attention of Dr. Craig for his opinion, I absolutely welcome such an endeavor!

mattdamore
03-10-2017, 10:34 AM
Oh my goodness, what is going on? This new website format is throwing me for a loop! Is anyone else having trouble? I was about to start my response and I don't know how to quote now! ha, ha.

shunyadragon
03-10-2017, 11:29 AM
Oh my goodness, what is going on? This new website format is throwing me for a loop! Is anyone else having trouble? I was about to start my response and I don't know how to quote now! ha, ha.

I am not presently having trouble.

How to quote:
La de da da [/quote ]

I added a space after the second quote. Take it out and it will read; [quote=Me] La de da da

Citation follows the same format; La de da da [/cite ]

I added a space after the second cite. Take it out and it will read; [cite=Me] La de da da

mattdamore
03-11-2017, 10:18 PM
I am not presently having trouble.

How to quote: [quote=Me] La de da da [/quote ]

I added a space after the second quote. Take it out and it will read;

Citation follows the same format; La de da da [/cite ]

I added a space after the second cite. Take it out and it will read; [cite=Me] La de da da

Thank you Shunya. I did already know that. But when I used to click on "go advanced", it would take me to a screen that had a whole panoply of options without the rigmarole of including the HTML coding manually. Did it change for you too? This would make it much more difficult for me to write responses. There used to be a veritable console of different tools that would automatically assist in providing HTML coding. I'm saddened that it's no longer here, unless my computer is wigging out.

mattdamore
03-11-2017, 10:20 PM
Oh my goodness! I fixed it! Yay! I went to the bottom of the screen and clicked a link called "see full site", or some such locution. I'm so sorry. Ignore the above post. I feel foolish.

Boxing Pythagoras
03-12-2017, 08:17 AM
Oh my goodness! I fixed it! Yay! I went to the bottom of the screen and clicked a link called "see full site", or some such locution. I'm so sorry. Ignore the above post. I feel foolish.
No worries! Happens to the best of us, from time to time.

mattdamore
10-05-2017, 08:22 AM
No worries! Happens to the best of us, from time to time.

Hello! It's been a long time! I hope you're doing well. I haven't forgotten about this conversation, thank goodness. Dr. Craig's class on God and Abstract objects starts Monday, and I'm excited to ask him about hyperreal numbers and infinity. If you'd like, perhaps you can type your question here, and I could read it to him in class. This is because I don't want to misrepresent you, ha. If I remember correctly, hyperreal numbers enable me to perform subtraction/division with transfinite numbers without paradox, and therefore serves as a sort of undercutting defeater to the thesis that an actual infinite is metaphysically impossible due to the paradoxes its instantiation in reality would lead to. Please let me know of any misrepresentation. Thanks!

shunyadragon
10-05-2017, 03:50 PM
Hello! It's been a long time! I hope you're doing well. I haven't forgotten about this conversation, thank goodness. Dr. Craig's class on God and Abstract objects starts Monday, and I'm excited to ask him about hyperreal numbers and infinity. If you'd like, perhaps you can type your question here, and I could read it to him in class. This is because I don't want to misrepresent you, ha. If I remember correctly, hyperreal numbers enable me to perform subtraction/division with transfinite numbers without paradox, and therefore serves as a sort of undercutting defeater to the thesis that an actual infinite is metaphysically impossible due to the paradoxes its instantiation in reality would lead to. Please let me know of any misrepresentation. Thanks!

The problem with considering actual infinities impossible is that they used to describe actual natural infinite sets in nature.



Do infinities exist in nature?
By Marianne Freiberger and Rachel Thomas

Submitted by Marianne on September 26, 2013
What would you see if you came to the edge of the Universe? It's hard to imagine so it's tempting to conclude that the Universe doesn't have an edge and therefore that it must be infinite. That's not a necessary conclusion however. There are things that are finite in extent but still don't have an edge, the prime example being the surface of a sphere. It's got a finite area but when you walk around on it you'll never fall over an edge. The question of whether the Universe is finite or infinite is one that still hasn't been answered, and there are mathematical models that allow for both possibilities. More generally, the question of whether any infinite quantities can arise in the Universe is a deep one. In April this year philosophers, cosmologists and physicists came together at the University of Cambridge, as part of a conference series on the philosophy of cosmology, in order to discuss it. Plus went along to find out more (and you can also listen to the interviews we did in our podcast).

Infinity that doesn't bite

John D. Barrow

People have been studying infinity and its relation to reality for a long time. "The idea of studying infinities in physics really began with Aristotle," says the Cambridge cosmologist John D. Barrow. "Aristotle made a clear distinction between two types of infinity. One he called potential infinities and he was quite happy to allow for those to appear in descriptions of the world. These are just like lists that never end. The ordinary numbers are an example; one, two, three, four, five, and so on, the list goes on forever. It's infinite, but you never reach or experience the infinity. In a subject like cosmology, there are lots of infinities like that and most people are quite happy with them. For example, the Universe might have infinite size; it might have an infinite past age, it might be destined to have an infinite future age. These are all potential infinities, so they don't bite you as it were, they're just ways of saying that things are limitless, they're unbounded, like that list of numbers."

mattbballman31
11-21-2017, 10:46 AM
The problem with considering actual infinities impossible is that they used to describe actual natural infinite sets in nature.

I feel like I'm going to regret responding to you, but I can't resist . . . sorry. So . . . you're claim is:

- Actual infinities are used to describe "actual natural infinite sets in nature".

. . . I have no idea what it means for an actual infinite set to be natural.



It's hard to imagine so it's tempting to conclude that the Universe doesn't have an edge and therefore that it must be infinite. That's not a necessary conclusion however.

How the heck doesn't this contradict what you just said? I'll keep reading . . .


The question of whether the Universe is finite or infinite is one that still hasn't been answered, and there are mathematical models that allow for both possibilities.

Irrelevant. Actual infinities are mathematically possible. No one denies this. Metaphysical possibility is the issue.


"Aristotle made a clear distinction between two types of infinity. One he called potential infinities and he was quite happy to allow for those to appear in descriptions of the world. These are just like lists that never end.

Wow. No one denies that potential infinities are real. Irrelevant.

So . . . . . . . . . . nothing you quoted supports your claim. I thought in my time off here you'd get a little smarter. Guess not.

shunyadragon
11-24-2017, 04:09 AM
I feel like I'm going to regret responding to you, but I can't resist . . . sorry. So . . . you're claim is:

- Actual infinities are used to describe "actual natural infinite sets in nature".

. . . I have no idea what it means for an actual infinite set to be natural.



How the heck doesn't this contradict what you just said? I'll keep reading . . .



Irrelevant. Actual infinities are mathematically possible. No one denies this. Metaphysical possibility is the issue.



Wow. No one denies that potential infinities are real. Irrelevant.

So . . . . . . . . . . nothing you quoted supports your claim. I thought in my time off here you'd get a little smarter. Guess not.

Metaphysical or physical? I believe you are misrepresenting Craig's argument.

Craig claims this by claiming our universe (physical existence) cannot be past infinite.



Craig on the actual infinite
wes morriston
Department of Philosophy, University of Colorado at Boulder, 169 Hellems, Campus
Box 232, Boulder, CO 80309-0232
Abstract : In a series of much discussed articles and books, William Lane Craig
defends the view that the past could not consist in a beginningless series of events.
In the present paper, I cast a critical eye on just one part of Craig’s case for the
finitude of the past – viz. his philosophical argument against the possibility of
actually infinite sets of objects in the ‘real world’. I shall try to show that this
argument is unsuccessful. I shall also take a close look at several considerations that
are often thought to favour the possibility of an actual infinite, arguing in each case
that Craig’s response is inadequate.

Craig's problem with infinities gets worse when he proposes the bogus old Hilbert's Hotel argument.


William Lane Craig explains why the universe had a beginning and is not infinite using the Hilbert's Hotel example. Since the universe had a beginning, it had to have a cause. That cause had to be enormously powerful and extremely intelligent, i.e. God.Feb 19, 2010
William Lane Craig explains Hilbert's Hotel, Infintiy, Kalam - YouTube
https://www.youtube.com/watch?v=j_q802eboxA

One easy response is simply begin with an infinite Hilbert's Hotel that is empty. Now, try and fill the hotel.

mattbballman31
11-24-2017, 08:19 PM
Metaphysical or physical? I believe you are misrepresenting Craig's argument.

Well, then you've been hating on him too much to know that I'm not.


Craig claims this by claiming our universe (physical existence) cannot be past infinite.


And how does this contradict the point that the past-eternal, physical existence of the universe is a metaphysical impossibility?


Abstract : In a series of much discussed articles and books, William Lane Craig
defends the view that the past could not consist in a beginningless series of events.
In the present paper, I cast a critical eye on just one part of Craig’s case for the
finitude of the past – viz. his philosophical argument against the possibility of
actually infinite sets of objects in the ‘real world’. I shall try to show that this
argument is unsuccessful. I shall also take a close look at several considerations that
are often thought to favour the possibility of an actual infinite, arguing in each case
that Craig’s response is inadequate.

See the bold stuff? That's referring to metaphysical possibility. That has absolutely nothing to do with whether the universe is physical or not, :lol:


Craig's problem with infinities gets worse when he proposes the bogus old Hilbert's Hotel argument.


Well, thank goodness I don't just take your word for it. Why is it bogus? Sounds good to me!


One easy response is simply begin with an infinite Hilbert's Hotel that is empty. Now, try and fill the hotel.

That was perspicuous, lol. The response is completely misconceived. Great, you've emptied it out. In this possible world, Craig's metaphysical absurdities involving contradictions in transfinite arithmetic wouldn't arise with regard to people leaving and accommodating a potentially infinite amount of new guests when an actual infinite amount of guests are already in the hotel. In the new scenario, the hotel would never fill up! That's Craig's second philosophical argument: the metaphysical impossibility of forming an actual infinite via successive addition. Thanks for illustrating Hilbert's Hotel in a way that doesn't threaten Craig's argument at all!

shunyadragon
11-27-2017, 05:43 PM
Well, then you've been hating on him too much to know that I'm not.



And how does this contradict the point that the past-eternal, physical existence of the universe is a metaphysical impossibility?


See the bold stuff? That's referring to metaphysical possibility. That has absolutely nothing to do with whether the universe is physical or not, :lol:



Well, thank goodness I don't just take your word for it. Why is it bogus? Sounds good to me!



That was perspicuous, lol. The response is completely misconceived. Great, you've emptied it out. In this possible world, Craig's metaphysical absurdities involving contradictions in transfinite arithmetic wouldn't arise with regard to people leaving and accommodating a potentially infinite amount of new guests when an actual infinite amount of guests are already in the hotel. In the new scenario, the hotel would never fill up! That's Craig's second philosophical argument: the metaphysical impossibility of forming an actual infinite via successive addition. Thanks for illustrating Hilbert's Hotel in a way that doesn't threaten Craig's argument at all!

I do not propose emptying anything. I propose beginning with a potentially infinite hotel that is empty. Now try and fill it. Hilbert's Hotel fits in my hotel.

Craig proposes that our (universe) physical existence cannot be 'past infinite.' That is not metaphysical issue.

Tassman
11-27-2017, 07:59 PM
And how does this contradict the point that the past-eternal, physical existence of the universe is a metaphysical impossibility?




Given you acknowledge that actual infinities are mathematically possible metaphysical possibility is no longer the issue. A metaphysician can be mistaken in his deductions, just as a scientist can. But even if these are impeccable, he will not necessarily succeed if he argues correctly from premises that are unacceptable because they lack the necessary foundation in fact.

mattbballman31
02-15-2018, 02:43 PM
I do not propose emptying anything. I propose beginning with a potentially infinite hotel that is empty. Now try and fill it. Hilbert's Hotel fits in my hotel.

Craig would have no problem saying you can't fill a hotel with a potentially infinite amount of rooms. His philosophical arguements don't even apply to a hotel with a potentially infintine amount of rooms in a hotel. What are you talking about?


Craig proposes that our (universe) physical existence cannot be 'past infinite.' That is not metaphysical issue.

Yes it is. Seriously. How is that not a metaphysical issue? The claims are,

1. An infinite temporal regress of events is an actual infinite
2. An actual infinite cannot exist.
3. Therefore, an infinite temporal regress of events cannot exist.

1 and 2 are metaphysical propositions, and so is the conclusion.

mattbballman31
02-15-2018, 03:01 PM
Given you acknowledge that actual infinities are mathematically possible metaphysical possibility is no longer the issue. A metaphysician can be mistaken in his deductions, just as a scientist can. But even if these are impeccable, he will not necessarily succeed if he argues correctly from premises that are unacceptable because they lack the necessary foundation in fact.

I'm so lost right now. When Craig is saying that an actual infinite is mathematically possible, he saying that you shouldn't deny the "mathematical legitimacy to the actual infinite", like the intuitionists do. "Cantor's system and axiomatized set theory may be taken to be simply a universe of discourse, a mathematical system based on certain adopted axioms and conventions, which carries no ontological commitments." Ontological commitments is the modal realm of metaphysical possibility, or broad logical possibility, which is in terms of actualizability. Like David Hilbert said, "The infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought . . . The role that remains for the infinite to play is solely that of an idea." So, just because you think there's mathematical legitimacy to the idea of actual infinites in infinite set theory, it does not follow that it is metaphysically possible to put into one-to-one correspondence each member of such a set and a concrete or a Platonically conceived abstract object. Craig doesn't even believe mathematical objects exist. Craig says, " . . . it is open to the mutakallim to hold that while the actual infinite is a fruitful and consistent concept within the postulated universe of discourse, it cannot be transposed into the real world."

shunyadragon
02-17-2018, 03:34 PM
I'm so lost right now. When Craig is saying that an actual infinite is mathematically possible, he saying that you shouldn't deny the "mathematical legitimacy to the actual infinite", like the intuitionists do. "Cantor's system and axiomatized set theory may be taken to be simply a universe of discourse, a mathematical system based on certain adopted axioms and conventions, which carries no ontological commitments." Ontological commitments is the modal realm of metaphysical possibility, or broad logical possibility, which is in terms of actualizability. Like David Hilbert said, "The infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought . . . The role that remains for the infinite to play is solely that of an idea." So, just because you think there's mathematical legitimacy to the idea of actual infinites in infinite set theory, it does not follow that it is metaphysically possible to put into one-to-one correspondence each member of such a set and a concrete or a Platonically conceived abstract object. Craig doesn't even believe mathematical objects exist. Craig says, " . . . it is open to the mutakallim to hold that while the actual infinite is a fruitful and consistent concept within the postulated universe of discourse, it cannot be transposed into the real world."

As Tassman noted, William Craig and David Hilbert may be mistaken in their deductions. Actual infinities are actually used in science to describe the physical nature of our existence. Actual infinities would descriptive of attributes of our physical existence within a greater potential infinity.

shunyadragon
02-17-2018, 03:39 PM
Craig would have no problem saying you can't fill a hotel with a potentially infinite amount of rooms. His philosophical arguments don't even apply to a hotel with a potentially infintine amount of rooms in a hotel. What are you talking about?

Potential infinities would only exist as infinite sets within a greater potential infinity.




Yes it is. Seriously. How is that not a metaphysical issue? The claims are,

1. An infinite temporal regress of events is an actual infinite
2. An actual infinite cannot exist.
3. Therefore, an infinite temporal regress of events cannot exist.

1 and 2 are metaphysical propositions, and so is the conclusion.

No, they describe limits on the potential infinite nature of our physical existence as Craig proposes. Actually the claim that an actual infinity cannot exist is false.

mattbballman31
02-18-2018, 01:41 PM
Potential infinities would only exist as infinite sets within a greater potential infinity.

Uh, potential infinities are finite sets.



No, they describe limits on the potential infinite nature of our physical existence as Craig proposes. Actually the claim that an actual infinity cannot exist is false.

Uh, yes. I don't care what they describe. I care about the nature of their descriptions. And your last point is completely irrelevant to that issue. Great! You think they can't exist. I disagree. But to say they can't exist is a metaphysical proposition about the impossibility of an actual infinite. The fact that you can't understand these simple distinctions means you just haven't read or understood the literature on the topic.

shunyadragon
02-18-2018, 02:00 PM
Uh, potential infinities are finite sets.


Actual infinities are defined as 'completed' closed sets of infinities.


For purposes of doing mathematics and science, the actual infinite has turned out to be the most useful of the three concepts. Using the idea proposed by Bolzano that was mentioned above, the concept of the actual infinite was precisely defined in 1888 when Richard Dedekind redefined the term “infinity” for use in set theory and George Cantor made the infinite, in the sense of infinite set, an object of mathematical study. Before this turning point, the philosophical community generally believed Aristotle’s concept of potential infinity should be the concept used in mathematics and science.


Completed infinity, or actual infinity, is an infinity that one actually reaches; the process is already done. For instance, let's put braces around that sequence mentioned earlier:
{ 1, 2, 3, 4, ... }
With this notation, we are indicating the set of all positive integers. This is just one object, a set. But that set has infinitely many members. By that I don't mean that it has a large finite number of members and it keeps getting more members. Rather, I mean that it already has infinitely many members.

We can also indicate the completed infinity geometrically. For instance, the diagram at right shows a one-to-one correspondence between points on an infinitely long line and points on a semicircle. (see reference for diagram) There are no points for plus or minus infinity on the line, but it is natural to attach those "numbers" to the endpoints of the semicircle.

Isn't that "cheating," to simply add numbers in this fashion? Not really; it just depends on what we want to use those numbers for. For instance, f(x)=1/(1+x2) is a continuous function defined for all real numbers x, and it also tends to a limit of 0 when x "goes to" plus or minus infinity (in the sense of potential infinity, described earlier). Consequently, if we add those two "numbers" to the real line, to get the so-called "extended real line," and we equip that set with the same topology as that of the closed semicircle (i.e., the semicircle including the endpoints), then the function f is continuous everywhere on the extended real line. This has some advantages in advanced mathematics: The topology of the closed semicircle is compact and metrizable. Compact metric spaces have very nice topological properties; for instance, every sequence has a convergent subsequence. Even if we're really just interested in the properties of the ordinary (finite) real numbers, we can discover and prove some of those properties more easily by viewing that set of numbers as a subset of this larger, compact metric space.




Uh, yes. I don't care what they describe. I care about the nature of their descriptions. And your last point is completely irrelevant to that issue. Great! You think they can't exist. I disagree. But to say they can't exist is a metaphysical proposition about the impossibility of an actual infinite. The fact that you can't understand these simple distinctions means you just haven't read or understood the literature on the topic.

Read my posts again and respond intelligently. Math is descriptive of our physical nature and math is not material objects that exist nor not exist. Actual infinities exists as descriptions of the physical properties of our physical existence.

If your care what they describe, they actually do exist as descriptions of attributes of our physical universe.



Do infinities exist in nature?
By
Marianne Freiberger and Rachel Thomas
Submitted by Marianne on September 26, 2013
What would you see if you came to the edge of the Universe? It's hard to imagine so it's tempting to conclude that the Universe doesn't have an edge and therefore that it must be infinite. That's not a necessary conclusion however. There are things that are finite in extent but still don't have an edge, the prime example being the surface of a sphere. It's got a finite area but when you walk around on it you'll never fall over an edge. The question of whether the Universe is finite or infinite is one that still hasn't been answered, and there are mathematical models that allow for both possibilities. More generally, the question of whether any infinite quantities can arise in the Universe is a deep one. In April this year philosophers, cosmologists and physicists came together at the University of Cambridge, as part of a conference series on the philosophy of cosmology, in order to discuss it. Plus went along to find out more (and you can also listen to the interviews we did in our podcast).

Infinity that doesn't bite
John D. Barrow

People have been studying infinity and its relation to reality for a long time. "The idea of studying infinities in physics really began with Aristotle," says the Cambridge cosmologist John D. Barrow. "Aristotle made a clear distinction between two types of infinity. One he called potential infinities and he was quite happy to allow for those to appear in descriptions of the world. These are just like lists that never end. The ordinary numbers are an example; one, two, three, four, five, and so on, the list goes on forever. It's infinite, but you never reach or experience the infinity. In a subject like cosmology, there are lots of infinities like that and most people are quite happy with them. For example, the Universe might have infinite size; it might have an infinite past age, it might be destined to have an infinite future age. These are all potential infinities, so they don't bite you as it were, they're just ways of saying that things are limitless, they're unbounded, like that list of numbers."

mattbballman31
02-18-2018, 02:39 PM
Actual infinities are defined as closed sets of infinities.

You really are dense.

YOU SAID THE POTENTIAL INFINITE WAS AN INFINITE SET, YOU MORON.

Then I said that the potential infinite was a finite set.

And your response is to mention that ACTUAL INFINITIES ARE DEFINED AS CLOSED SETS OF INFINITIES????

NO DUH! It's like I'm talking to a darn wall. I wasn't talking about actual infinites, moron. I was responding to your claim that potential infinites are infinite sets. They aren't. Got that, moron? So, your dumb IEP quote is idiotically irrelevant, moron. Learn to read and stop wasting my time.



Read my posts again and respond intelligently.

Why don't you follow your own genius advice? :lol:


Math is descriptive of our physical nature and math is not material objects that exist nor not exist. Actual infinities exists as descriptions of the physical properties of our physical existence.


Did you even read my last post, moron? I said that . . . Craig . . . does not deny the mathematical legitimacy of the actual infinite. THAT MEANS I DON'T DENY THAT MATH IS DESCRIPTIVE OF PHYSICAL NATURE, MORON. How in the freaking world does this have anything to do with the idea that " . . . it is open to the mutakallim to hold that while the actual infinite is a fruitful and consistent concept within the postulated universe of discourse, it cannot be transposed into the real world."

And then you just quote these huge block quotes I've already addressed, dunce. Marianne Freiberger and Rachel Thomas don't say ANYTHING addressing the point. And Barrow is only talking about POTENTIAL infinities. You either really suck at reading and comprehension or you're so blinded by your own bias that any semblance of objectivity has evaporated into the stratosphere and beyond.

I can't wait what nonsense you'll spew next! Stay tuned! :lol:

shunyadragon
02-18-2018, 04:37 PM
You really are dense.

YOU SAID THE POTENTIAL INFINITE WAS AN INFINITE SET, YOU MORON.

Then I said that the potential infinite was a finite set.

And your response is to mention that ACTUAL INFINITIES ARE DEFINED AS CLOSED SETS OF INFINITIES????

NO DUH! It's like I'm talking to a darn wall. I wasn't talking about actual infinites, moron. I was responding to your claim that potential infinites are infinite sets. They aren't. Got that, moron? So, your dumb IEP quote is idiotically irrelevant, moron. Learn to read and stop wasting my time.



Why don't you follow your own genius advice? :lol:



Did you even read my last post, moron? I said that . . . Craig . . . does not deny the mathematical legitimacy of the actual infinite. THAT MEANS I DON'T DENY THAT MATH IS DESCRIPTIVE OF PHYSICAL NATURE, MORON. How in the freaking world does this have anything to do with the idea that " . . . it is open to the mutakallim to hold that while the actual infinite is a fruitful and consistent concept within the postulated universe of discourse, it cannot be transposed into the real world."

And then you just quote these huge block quotes I've already addressed, dunce. Marianne Freiberger and Rachel Thomas don't say ANYTHING addressing the point. And Barrow is only talking about POTENTIAL infinities. You either really suck at reading and comprehension or you're so blinded by your own bias that any semblance of objectivity has evaporated into the stratosphere and beyond.

I can't wait what nonsense you'll spew next! Stay tuned! :lol:

I apologize for the typo. It is actual infinities that are 'complete' closed sets. They are description of attributes of our physical existence.

I correctly described it in the next post. In this context actual infinities do indeed exist in nature.

mattbballman31
02-18-2018, 07:52 PM
I apologize for the typo. It is actual infinities that are 'complete' closed sets. They are description of attributes of our physical existence.

I correctly described it in the next post. In this context actual infinities do indeed exist in nature.

Wow. Admission of error. We're making progress! There's hope for you yet.

The quote you gave doesn't show that actual infinities exist in nature at all. It's like you copy/past some mathematical explanation and are unable to connect the dots to metaphysical reality. All the quote does is define, mathematically, an actual infinite (something Craig affirms), extend such a definition geometrically from points on a line to points on a semicircle (something Craig wouldn't deny), and show how it wouldn't be 'cheating' if such a topological extension is done in geometry. Wow. Cool. Oooh! Ahhh! Shuny can quote blocks of irrelevant bilge. Craig denies none of this, and it has nothing (absolutely nothing!) to do with saying that actual infinities exist in nature.

Oh, and because you're such a Craig-expert, quote me exactly where Craig addresses the very point your clumsily trying to target that because one can successfully utilize actual infinites relative to topological geometrical structures for whatever the mathematician 'wants to use the numbers for', that this therefore implies the begged metaphysical question of whether the topological structure of space itself is metaphysically identical to the structure as delineated in geometry. He addresses it! It's in his literature on the topic! And rather than hold your hand and quote it for you, either admit that you don't have a sufficient grasp on the argument as Craig has presented it since freaking 1979 or that you're an embarrassing armchair block-quote copy and paster whose only concern is the appearance of sophistication, an appearance that itself is a laughable disguise. I'm so sick of pseudo-intellectuals like you thinking that you can just throw some stupid gotchya-spaghetti against the wall, and you actually are convinced that you've GOT CRAIG WITH HIS BACK AGAINST THE WALL. It's absurd. He's explored the darn argument almost into the ground, and these silly 'arguments' you bring up would just be laughed out of your normal philosophy conference.

shunyadragon
02-19-2018, 08:11 AM
This article by Morriston addresses the problems with Craig's arguments concerning actual infinities in great detail.




In a series of much discussed articles and books, William Lane Craig has
vigorously defended the view that the past could not consist in a beginningless
series of events.1 Craig’s goal, of course, is to make a strong case for the existence
of God. If the past has a beginning, then so does the universe, and a familiar line
of argument suggests that there must be a First Cause.2 In the present paper, I cast
a critical eye on just one part of Craig’s case for the finitude of the past – viz. his
philosophical argument against the possibility of actually infinite sets in the ‘real
world’.3 If this argument were to succeed, then an actually infinite series of past
events would have been proved impossible, and we could go on to ask about the
cause of the very first event. However, I do not believe that Craig has succeeded in
proving that actually infinite sets are impossible. As far as this particular line of
argument is concerned, I shall try to show that it remains an open question
whether the past could consist in a beginningless series of events. I shall also take a
close look at several considerations that are often thought to favour the possibility
of an actual infinite, arguing in each case that Craig’s response is inadequate.

My previous references confirm that actual infinities do exist in nature.

Chrawnus
02-19-2018, 09:24 AM
My previous references confirm that actual infinities do exist in nature.

No they didn't. I read through the whole article you referenced in post #53, and not once did I see the authors make the claim that actual infinities have been confirmed to exist in nature.

shunyadragon
02-19-2018, 07:29 PM
No they didn't. I read through the whole article you referenced in post #53, and not once did I see the authors make the claim that actual infinities have been confirmed to exist in nature.

Yes it demonstrated Craig's assertions as false, as well as other references describing actual infinities in nature. More to follow.

Self imposed ignorance leers its ugly head.

Again this article discusses actual infinities as they exist in describing their properties using math to describe nature.


Does infinity exist? by John D. Barrow
Submitted by Marianne on July 2, 2012

In the latest poll of our Science fiction, science fact project you told us that you wanted to know if infinity exists. Here is an answer, based on an interview with the cosmologist John D. Barrow. Click here to see other articles on infinity and here to listen to our interview with Barrow as a podcast.

. . .


But generally Cantor's ideas have been accepted and today they form their own sub-branch of pure mathematics. This has led some philosophers, and even some theologians, to rethink their ancient attitudes to infinities. Because there are quite different varieties of infinity, it is clear that you don't have to regard the appearance of mathematical infinity as some sort of challenge to the divine as the medieval theologians believed. Cantor's ideas were at first actually taken up more enthusiastically by contemporay theologians than by mathematicians.

Scientists also started to distinguish between mathematical and physical infinities. In mathematics, if you say something "exists", what you mean is that it doesn't introduce a logical contradiction given a particular set of rules. But it doesn't mean that you can have one sitting on your desk or that there's one running around somewhere. Unicorns are not a logical impossibility but that doesn't mean that one exists biologically. When mathematicians demonstrated that non-Euclidean geometries can exist, they showed that there's an axiomatic system which permits them that is not self-contradictory. (You can find out more about non-Euclidean geometries in the article Strange geometries.)

Physical infinities
So infinities in modern physics have become separate from the study of infinities in mathematics. One area in physics where infinities are sometimes predicted to arise is aerodynamics or fluid mechanics. For example, you might have a wave becoming very, very steep and non-linear and then forming a shock. In the equations that describe the shock wave formation some quantities may become infinite. But when this happens you usually assume that it's just a failure of your model. You might have neglected to take account of friction or viscosity and once you include that into your equations the velocity gradient becomes finite — it might still be very steep, but the viscosity smoothes over the infinity in reality. In most areas of science, if you see an infinity, you assume that it's down to an inaccuracy or incompleteness of your model.

string diagram
Two particles meeting form a sharp corner (left) but two loops coming together are like two pairs of trousers sown together. (The trouser diagram has time going downwards and space horizontal.)

In particle physics there has been a much longer-standing and more subtle problem. Quantum electrodynamics is the best theory in the whole of science, its predictions are more accurate than anything else that we know about the Universe. Yet extracting those predictions presented an awkward problem: when you did a calculation to see what you should observe in an experiment you always seemed to get an infinite answer with an extra finite bit added on. If you then subtracted off the infinity, the finite part that you were left with was the prediction you expected to see in the lab. And this always matched experiment fantastically accurately. This process of removing the infinities was called renormalisation. Many famous physicists found it deeply unsatisfactory. They thought it might just be a symptom of a theory that could be improved.

This is why string theory created great excitement in the 1980s and why it suddenly became investigated by a huge number of physicists. It was the first time that particle physicists found a finite theory, a theory which didn't have these infinities popping up. The way it did it was to replace the traditional notion that the most basic entities in the theory (for example photons or electrons) should be point-like objects that move through space and time and so trace out lines in spacetime. Instead, string theory considers the most basic entities to be lines, or little loops, which trace out tubes as they move. When you have two point-like particles moving through space and interacting, it's like two lines hitting one another and forming a sharp corner at the place where they meet. It's that sharp corner in the picture that's the source of the infinities in the description. But if you have two loops coming together, it's rather like two legs of a pair of trousers. Then two more loops move out from the interaction — that's like sewing another pair of trousers onto the first pair. What you get is a smooth transition. This was the reason why string theory was so appealing, it was the first finite theory of particle physics.

Cosmological infinities
Black hole
Simulated view of a black hole. Image: Alain Riazuelo.

Another type of infinity arises in gravitation theory and cosmology. Einstein's theory of general relativity suggests that an expanding Universe (as we observe ours to be) started at a time in the finite past when its density was infinite — this is what we call the Big Bang. Einstein's theory also predicts that if you fell into a black hole, and there are many black holes in our Galaxy and nearby, you would encounter an infinite density at the centre. These infinities, if they do exist, would be actual infinities.

Chrawnus
02-19-2018, 08:50 PM
Yes it demonstrated Craig's assertions as false, as well as other references describing actual infinities in nature. More to follow.

Self imposed ignorance leers its ugly head.

Again this article discusses actual infinities as they exist in describing their properties using math to describe nature.


Does infinity exist? by John D. Barrow
Submitted by Marianne on July 2, 2012

In the latest poll of our Science fiction, science fact project you told us that you wanted to know if infinity exists. Here is an answer, based on an interview with the cosmologist John D. Barrow. Click here to see other articles on infinity and here to listen to our interview with Barrow as a podcast.

. . .


But generally Cantor's ideas have been accepted and today they form their own sub-branch of pure mathematics. This has led some philosophers, and even some theologians, to rethink their ancient attitudes to infinities. Because there are quite different varieties of infinity, it is clear that you don't have to regard the appearance of mathematical infinity as some sort of challenge to the divine as the medieval theologians believed. Cantor's ideas were at first actually taken up more enthusiastically by contemporay theologians than by mathematicians.

Scientists also started to distinguish between mathematical and physical infinities. In mathematics, if you say something "exists", what you mean is that it doesn't introduce a logical contradiction given a particular set of rules. But it doesn't mean that you can have one sitting on your desk or that there's one running around somewhere. Unicorns are not a logical impossibility but that doesn't mean that one exists biologically. When mathematicians demonstrated that non-Euclidean geometries can exist, they showed that there's an axiomatic system which permits them that is not self-contradictory. (You can find out more about non-Euclidean geometries in the article Strange geometries.)

Physical infinities
So infinities in modern physics have become separate from the study of infinities in mathematics. One area in physics where infinities are sometimes predicted to arise is aerodynamics or fluid mechanics. For example, you might have a wave becoming very, very steep and non-linear and then forming a shock. In the equations that describe the shock wave formation some quantities may become infinite. But when this happens you usually assume that it's just a failure of your model. You might have neglected to take account of friction or viscosity and once you include that into your equations the velocity gradient becomes finite — it might still be very steep, but the viscosity smoothes over the infinity in reality. In most areas of science, if you see an infinity, you assume that it's down to an inaccuracy or incompleteness of your model.

string diagram
Two particles meeting form a sharp corner (left) but two loops coming together are like two pairs of trousers sown together. (The trouser diagram has time going downwards and space horizontal.)

In particle physics there has been a much longer-standing and more subtle problem. Quantum electrodynamics is the best theory in the whole of science, its predictions are more accurate than anything else that we know about the Universe. Yet extracting those predictions presented an awkward problem: when you did a calculation to see what you should observe in an experiment you always seemed to get an infinite answer with an extra finite bit added on. If you then subtracted off the infinity, the finite part that you were left with was the prediction you expected to see in the lab. And this always matched experiment fantastically accurately. This process of removing the infinities was called renormalisation. Many famous physicists found it deeply unsatisfactory. They thought it might just be a symptom of a theory that could be improved.

This is why string theory created great excitement in the 1980s and why it suddenly became investigated by a huge number of physicists. It was the first time that particle physicists found a finite theory, a theory which didn't have these infinities popping up. The way it did it was to replace the traditional notion that the most basic entities in the theory (for example photons or electrons) should be point-like objects that move through space and time and so trace out lines in spacetime. Instead, string theory considers the most basic entities to be lines, or little loops, which trace out tubes as they move. When you have two point-like particles moving through space and interacting, it's like two lines hitting one another and forming a sharp corner at the place where they meet. It's that sharp corner in the picture that's the source of the infinities in the description. But if you have two loops coming together, it's rather like two legs of a pair of trousers. Then two more loops move out from the interaction — that's like sewing another pair of trousers onto the first pair. What you get is a smooth transition. This was the reason why string theory was so appealing, it was the first finite theory of particle physics.

Cosmological infinities
Black hole
Simulated view of a black hole. Image: Alain Riazuelo.

Another type of infinity arises in gravitation theory and cosmology. Einstein's theory of general relativity suggests that an expanding Universe (as we observe ours to be) started at a time in the finite past when its density was infinite — this is what we call the Big Bang. Einstein's theory also predicts that if you fell into a black hole, and there are many black holes in our Galaxy and nearby, you would encounter an infinite density at the centre. These infinities, if they do exist, would be actual infinities.

Can you please bold the part that you misunderstood to confirm the existence of actual infinities in nature?

shunyadragon
02-20-2018, 02:05 PM
Can you please bold the part that you misunderstood to confirm the existence of actual infinities in nature?

I do not spoon feed the intentional ignorant. The references speaks for themself. I detect a severe deficiency in your English and math comprehension?

mattbballman31
02-20-2018, 02:10 PM
Yippee!! More hurling elephantine block-quotes to choke on! Let's see if they're just more inconsequential poppycock. It must be nice to do absolutely no work going through the time to actually understand the issues. It's the dark side of the Internet. Let me just put "William Lane Craig", "Infidels", and "Morriston" into Google, cross my fingers and toes, close my eyes, click my heels, control+F 'actual infinite', and copy/paste a big, shiny block-paragraph for my response. :rofl:


In a series of much discussed articles and books, William Lane Craig has
vigorously defended the view that the past could not consist in a beginningless
series of events.1 Craig’s goal, of course, is to make a strong case for the existence
of God. If the past has a beginning, then so does the universe, and a familiar line
of argument suggests that there must be a First Cause.2 In the present paper, I cast
a critical eye on just one part of Craig’s case for the finitude of the past – viz. his
philosophical argument against the possibility of actually infinite sets in the ‘real
world’.3 If this argument were to succeed, then an actually infinite series of past
events would have been proved impossible, and we could go on to ask about the
cause of the very first event. However, I do not believe that Craig has succeeded in
proving that actually infinite sets are impossible. As far as this particular line of
argument is concerned, I shall try to show that it remains an open question
whether the past could consist in a beginningless series of events. I shall also take a
close look at several considerations that are often thought to favour the possibility
of an actual infinite, arguing in each case that Craig’s response is inadequate. [/cite]

My previous references confirm that actual infinities do exist in nature.

Prophecy fulfilled! I love being right. Thanks for quoting a darn 'abstract' addressing absolutely nothing I said. Have you read Craig's response to Morriston, or do you have a Morriston-statue by your bedside you pray to before you go to sleep every night? :yipee:

mattbballman31
02-20-2018, 02:14 PM
I do not spoon feed the intentional ignorant. The references speaks for themself. I detect a severe deficiency in your English and math comprehension?

:lol::lol::lol::lol::lol::lol::lol:

I have to call the Guinness Book of World Records and let them know that this might be the blackest pot of all time calling a kettle that's not black 'black'! :rofl:

shunyadragon
02-20-2018, 02:40 PM
:lol::lol::lol::lol::lol::lol::lol:

I have to call the Guinness Book of World Records and let them know that this might be the blackest pot of all time calling a kettle that's not black 'black'! :rofl:

I do not spoon feed the intentional ignorant. The references speaks for themself. I detect a severe deficiency in your English and math comprehension?

Smiley faces do not help your case.

mattbballman31
02-20-2018, 04:03 PM
I do not spoon feed the intentional ignorant.

Funny. You gobbled up the spoon I was feeding you with a long time ago. Now, you just puke up extraneous block-quotes that are as irrelevant as they are desperate. Your misunderstand-Craig-derangement syndrome knows no bounds. :lol:


The references speaks for themself.


Nice English skills, dolt! :lol: 'Themself'? :rofl:


I detect a severe deficiency in your English and math comprehension?


Your detection device got darn broke in yer Dunning-Krueger algorithm. Seriously. For you to talk about someone's English comprehension is freaking hilarious. Your math comprehension is equally hilarious. Just nodding, drooling in the corner, at whatever beneficial, constructive points Boxing Pythagoras had made earlier in the thread does not a competent math guy make.


Smiley faces do not help your case.

I know, but aren't they so cute??? :teeth:

mattbballman31
02-20-2018, 04:49 PM
Let me slog through your probably irrelevant Barrow quotation from the 10th circle of Hell Dante didn't mention because he didn't have the misfortune to talk to you on this forum.


Scientists also started to distinguish between mathematical and physical infinities. In mathematics, if you say something "exists", what you mean is that it doesn't introduce a logical contradiction given a particular set of rules.

Cool. Craig addresses this. So scared! I would tell you what he says, but I don't spoon feed the intentionally ignorant! :lol:


In most areas of science, if you see an infinity, you assume that it's down to an inaccuracy or incompleteness of your model.


:rofl: So, no infinity in nature. Thanks, shuny!


when you did a calculation to see what you should observe in an experiment you always seemed to get an infinite answer with an extra finite bit added on. If you then subtracted off the infinity, the finite part that you were left with was the prediction you expected to see in the lab. And this always matched experiment fantastically accurately.

Calculation? Right, dimwit. Calculation. Craig doesn't deny the mathematical legitimacy of the actual infinite. And you saw the finite part "in the lab", in reality, in nature. Got that, weirdo?



This is why string theory created great excitement in the 1980s and why it suddenly became investigated by a huge number of physicists. It was the first time that particle physicists found a finite theory, a theory which didn't have these infinities popping up.

A theory that DIDN'T have the infinities popping up????? Do you even read these block-quotes before you paste them? :lol:


It's that sharp corner in the picture that's the source of the infinities in the description.


Here Shuny is drooling with excitement! Oh my gosh! Look! A block-quote with "infinities in the description" in it!!! Yippee! Oh wait, I have to read the next part of Barrow's quote, :yipee:


But if you have two loops coming together, it's rather like two legs of a pair of trousers. Then two more loops move out from the interaction — that's like sewing another pair of trousers onto the first pair. What you get is a smooth transition. This was the reason why string theory was so appealing, it was the first finite theory of particle physics.


See that, numbskull? "The first FINITE theory of particle physics"???? Your English comprehension is really atrocious. Nothing you quoted from Barrow supports the idea of infinities in nature, dummy.

What about this Alain Riazuelo?


Another type of infinity arises in gravitation theory and cosmology. Einstein's theory of general relativity suggests that an expanding Universe (as we observe ours to be) started at a time in the finite past when its density was infinite — this is what we call the Big Bang. Einstein's theory also predicts that if you fell into a black hole, and there are many black holes in our Galaxy and nearby, you would encounter an infinite density at the centre. These infinities, if they do exist, would be actual infinities.

You really are pea-brained. This is on the level of mathematical descriptions of densities in nature, which Craig would yawn at. Physicists, or pea-brained sycophants like you, that just dogmatically assert as settled the continuous nature of space (and therefore its infinite divisibility) are just clueless about the raging debate in physics, metaphysics and philosophy of science as to what the proper physical interpretations of the mathematical formalisms are regarding how quantum physics and general relativity imply that space is discrete or continuous, along with the debate swirling around whether you have to be a mathematical realist regarding the infinitesimals in the equations needing to correspond to discrete, mathematical quanta in the continuum. All of this is still up for debate, with physicists split down the middle, so maybe you shouldn't quote-mine to substantiate your confirmation bias. Oh, and Craig is aware of all this hooey already. He wrote a huge tome you probably won't read (and I won't tell you about, since I won't spoon feed the intentionally ignorant) about why the aforesaid mathematical realism is groundless and false. Further, more and more physicists are acknowledging the potency of the philosophical arguments from metaphysics as posing significant external conceptual problems for those physical interpretations of the formalism of general relativity and quantum mechanics that imply that an actual infinite of discrete quanta of space are logically prior to the continuous nature of space as a whole.

But I won't expect you to understand or read any of this. Go fetch another block-quote doggy! :lol:

Tassman
02-20-2018, 10:49 PM
Let me slog through your probably irrelevant Barrow quotation from the 10th circle of Hell Dante didn't mention because he didn't have the misfortune to talk to you on this forum.



Cool. Craig addresses this. So scared! I would tell you what he says, but I don't spoon feed the intentionally ignorant! :lol:



:rofl: So, no infinity in nature. Thanks, shuny!



Calculation? Right, dimwit. Calculation. Craig doesn't deny the mathematical legitimacy of the actual infinite. And you saw the finite part "in the lab", in reality, in nature. Got that, weirdo?



A theory that DIDN'T have the infinities popping up????? Do you even read these block-quotes before you paste them? :lol:



Here Shuny is drooling with excitement! Oh my gosh! Look! A block-quote with "infinities in the description" in it!!! Yippee! Oh wait, I have to read the next part of Barrow's quote, :yipee:



See that, numbskull? "The first FINITE theory of particle physics"???? Your English comprehension is really atrocious. Nothing you quoted from Barrow supports the idea of infinities in nature, dummy.

What about this Alain Riazuelo?



You really are pea-brained. This is on the level of mathematical descriptions of densities in nature, which Craig would yawn at. Physicists, or pea-brained sycophants like you, that just dogmatically assert as settled the continuous nature of space (and therefore its infinite divisibility) are just clueless about the raging debate in physics, metaphysics and philosophy of science as to what the proper physical interpretations of the mathematical formalisms are regarding how quantum physics and general relativity imply that space is discrete or continuous, along with the debate swirling around whether you have to be a mathematical realist regarding the infinitesimals in the equations needing to correspond to discrete, mathematical quanta in the continuum. All of this is still up for debate, with physicists split down the middle, so maybe you shouldn't quote-mine to substantiate your confirmation bias. Oh, and Craig is aware of all this hooey already. He wrote a huge tome you probably won't read (and I won't tell you about, since I won't spoon feed the intentionally ignorant) about why the aforesaid mathematical realism is groundless and false. Further, more and more physicists are acknowledging the potency of the philosophical arguments from metaphysics as posing significant external conceptual problems for those physical interpretations of the formalism of general relativity and quantum mechanics that imply that an actual infinite of discrete quanta of space are logically prior to the continuous nature of space as a whole.

But I won't expect you to understand or read any of this. Go fetch another block-quote doggy! :lol:

There is something seriously wrong with your attitude towards fellow members.

shunyadragon
02-21-2018, 04:39 AM
Let me slog through your probably irrelevant Barrow quotation from the 10th circle of Hell Dante didn't mention because he didn't have the misfortune to talk to you on this forum.



Cool. Craig addresses this. So scared! I would tell you what he says, but I don't spoon feed the intentionally ignorant! :lol:



:rofl: So, no infinity in nature. Thanks, shuny!



Calculation? Right, dimwit. Calculation. Craig doesn't deny the mathematical legitimacy of the actual infinite. And you saw the finite part "in the lab", in reality, in nature. Got that, weirdo?



A theory that DIDN'T have the infinities popping up????? Do you even read these block-quotes before you paste them? :lol:



Here Shuny is drooling with excitement! Oh my gosh! Look! A block-quote with "infinities in the description" in it!!! Yippee! Oh wait, I have to read the next part of Barrow's quote, :yipee:



See that, numbskull? "The first FINITE theory of particle physics"???? Your English comprehension is really atrocious. Nothing you quoted from Barrow supports the idea of infinities in nature, dummy.

What about this Alain Riazuelo?



You really are pea-brained. This is on the level of mathematical descriptions of densities in nature, which Craig would yawn at. Physicists, or pea-brained sycophants like you, that just dogmatically assert as settled the continuous nature of space (and therefore its infinite divisibility) are just clueless about the raging debate in physics, metaphysics and philosophy of science as to what the proper physical interpretations of the mathematical formalisms are regarding how quantum physics and general relativity imply that space is discrete or continuous, along with the debate swirling around whether you have to be a mathematical realist regarding the infinitesimals in the equations needing to correspond to discrete, mathematical quanta in the continuum. All of this is still up for debate, with physicists split down the middle, so maybe you shouldn't quote-mine to substantiate your confirmation bias. Oh, and Craig is aware of all this hooey already. He wrote a huge tome you probably won't read (and I won't tell you about, since I won't spoon feed the intentionally ignorant) about why the aforesaid mathematical realism is groundless and false. Further, more and more physicists are acknowledging the potency of the philosophical arguments from metaphysics as posing significant external conceptual problems for those physical interpretations of the formalism of general relativity and quantum mechanics that imply that an actual infinite of discrete quanta of space are logically prior to the continuous nature of space as a whole.

But I won't expect you to understand or read any of this. Go fetch another block-quote doggy! :lol:

Your selective citations to justify a religious agenda do not reflect the content of the whole reference. Smiley faces are a nice smoke screen for self imposed ignorance.

mattbballman31
02-21-2018, 06:01 PM
There is something seriously wrong with your attitude towards fellow members.

There's something seriously wrong with your face . . . :rofl:

mattbballman31
02-21-2018, 06:02 PM
Your selective citations to justify a religious agenda do not reflect the content of the whole reference. Smiley faces are a nice smoke screen for self imposed ignorance.

Okay, dummy. Whatever you say. Go get lost! :lol:

Tassman
02-21-2018, 09:00 PM
Okay, dummy. Whatever you say. Go get lost! :lol:

Gotta love the Christians!

Chrawnus
02-21-2018, 09:06 PM
I do not spoon feed the intentional ignorant.

Why should I do your job for you? :huh:

mattbballman31
02-22-2018, 05:15 PM
Gotta love the Christians!

We love you too, Tass Guy the Scientism Guy! :lol:

shunyadragon
03-27-2018, 05:21 AM
Why should I do your job for you? :huh:

Your the who asked me to do your laundry. It is your problem that you are not will to read the references and do the homework. At present you have an F-.

For example in the course of this thread Boxing Pythagoris does do the homework and responds intelligently to the thread when responding in the dialogue.




At this point, I need to use characters for the Greek alphabet, which I don't have. I'll use an English transliteration, but I apologize if this is cause for confusion. I'll try my best to be as clear as I can.



Most fonts support Greek and Hebrew characters, these days, but you'll have to use some extra manner of accessing them-- for example, the Charmap program in Windows or http://typegreek.com/

Another option, and the one which I prefer, is to use LaTeX formatted images. This tool is very helpful in that regard: http://www.codecogs.com/latex/eqneditor.php

Omega = Aleph null - [Omega is the first infinite ordinal. Aleph null is first infinite cardinal: the set of all natural numbers. The set of all natural numbers is countably infinite. - The sets above this point are all finite.]
It is not quite true that . The cardinality of omega is Aleph null, but ordinals and cardinals are very different sorts of numbers. We can't just equate them in this way. For example, it is true that ; however, we know that . Saying that these two numbers equal one another would make our mathematics inconsistent.



As I said, I am a rank amateur when it comes to this concept, and I have probably made mistakes above. Any contribution or clarification is most welcome.



In general, it is useful to note the difference between ordinal and cardinal numbers. Ordinals, as their name implies, are a description of how elements of a set may be ordered. Cardinals, on the other hand, are a description of how the elements of one set can be mapped onto another. So, ω describes the first number which is ordinally greater than any Natural number, in transfinite arithmetic. On the other hand, represents the cardinality of any set which can be mapped with 1-to-1 correspondence onto the Natural numbers.

I have consistently followed and agree with Boxing Pythagoris's view of Math concerning infinities/ Where are you at?



Calculus is a tool for calculating actually infinite sets of objects, and this tool can be used to describe the workings of the real world with incredible accuracy, as Newton demonstrated with his celestial mechanics. There is some philosophical debate, however, as to whether such tools are just decent idealizations which provide reasonable approximations of reality, or whether they accurately describe reality.

I'm a Formalist when it comes to the philosophy of mathematics, so I think ALL mathematics is a purely abstract means of describing the real world-- including basic arithmetic. However, there are ways in which this abstraction can be more or less accurate in its description of the world.

Actually, doing your own homework can amount to simply reading Boxing Pythagoris's posts.