# Thread: The Concept of the Infinite

1. Originally Posted by Boxing Pythagoras
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.

2. 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 $\emptyset$ 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 {$\emptyset$}.

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 {$\emptyset$, {$\emptyset$}}.

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.

3. Originally Posted by Boxing Pythagoras
"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.

4. Originally Posted by Boxing Pythagoras
Yep, the superscripted 2's imply power operations on their respective numbers. So, the square of $\aleph_0$ is equal to $\aleph_0$, 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 $\aleph_0$.
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 $\aleph_0$, 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.

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!

6. 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.

7. Originally Posted by mattdamore
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 $\aleph _0$, nonetheless we cannot say that $\omega=\aleph _0$. 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.

8. [QUOTE=mattdamore;418453]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?
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.

[quote] 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.

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 . . .

Source: https://en.wikipedia.org/wiki/Actual_infinity

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.

9. Originally Posted by mattdamore
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.

10. Originally Posted by Boxing Pythagoras
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.

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