Thread: The Concept of the Infinite

1. The Concept of the Infinite

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.

2. 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.

3. Originally Posted by Leonhard
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.

4. Originally Posted by mattdamore
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 $\omega=\aleph _0$. 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 $\aleph _0 =\aleph _0^2$; however, we know that $\omega < \omega^2$. 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, $\aleph_0$ represents the cardinality of any set which can be mapped with 1-to-1 correspondence onto the Natural numbers.

5. Originally Posted by Boxing Pythagoras
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 $\omega=\aleph _0$. 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 $\aleph _0 =\aleph _0^2$; however, we know that $\omega < \omega^2$. 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 $\aleph _0 =\aleph _0^2$? Is it raising $\aleph_0$ 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 $\omega < \omega^2$ 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, $\aleph_0$ 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.

6. Originally Posted by mattdamore
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 $\aleph _0 =\aleph _0^2$? Is it raising $\aleph_0$ 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 $\omega < \omega^2$ is the case, cardinality is distinct from ordinality.
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. 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 $\aleph_0$.

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 $\aleph_0$, 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.

7. 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?'

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.

9. 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.

10. Originally Posted by 37818
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.

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