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'sThe 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]

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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- [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.]_{null}

-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.]

-Omega- [These are limit ordinals, bigger than ordinals from infinite sets.]^{2}

-Omega- [This is the least ordinal that's bigger than all the natural powers of omega.]^{Omega}

-Omega- [It cannot be expressed by an infinite amount of natural powers of omega.]^{OmegaOmega}

-Epsilon- [Epsilon null. The first inaccessible ordinal.]_{0}

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

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

-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.