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I reckon the reasoning behind of some axioms in set theory. The axiom of extent is to say that we only care about what the sets contain, an abstraction for containers; the axiom of pair, the axioms of union, the axiom of power make sense for the construction of sets, from others; since the axiom of subsets is impossible to be proven by the others axioms, as I unsurely believe, then its assertion is essential. The axiom of regularity makes the felling of stupidity emanate from the reader, for its wonder.

But what in the hell is with the axiom of infinity? It seems like it only exists to say that the set of the natural numbers exists, so why it just does not assert that? My only is guess is (optional read):

that it is as if the axiom of infinity asserted infinity, and the way it does that we use to build the natural numbers; but, whilst making those axioms, the intention was for the existence of the natural numbers and the existence of infinity to be related, since the set of the natural numbers looks like the simplest infinity, and then we build from that, other infinity-things. That is, we assert the existence of infinity by asserting the existence of the simplest kind of infinity. So the axiom of infinity does not simply declare the existence of the set of the natural numbers because it has bigger plans: it asserts the existence of infinity using the natural numbers (asserting its existence), but it is not FOR the natural numbers (for asserting its existence).

Why is the axiom of infinity defined as it is, and has it any use besides proving the existence of the set of the natural numbers?

Schilive
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  • By the way, I am not sure this question is appropriate for this site, and I am defining the natural numbers on my book. So I make A LOT of guesses in my question. Since I dramatized it a little, it seems a bit dumber too. – Schilive May 28 '22 at 23:57
  • Did you read answers to this question? – Moishe Kohan May 29 '22 at 00:03
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    “It seems like it only exists to say that the set of the natural numbers exists, so why it just does not assert that?” It does say that. That is exactly what it says. But everything in ZF is a set, so you have to specify what it means to be a “natural number”. – Mark Saving May 29 '22 at 00:04
  • @MoisheKohan, I did, but I did not understand them. I did not seem them saying what the axiom of infinity tries to assert more specifically than *INFINITY IS NIGH*, whose truth was said to be debatable. – Schilive May 29 '22 at 00:10
  • @MarkSaving, perhaps, but it would be very simple to say $\exists_{x}\forall_{y}(y \in x \leftrightarrow [y = \emptyset \lor \exists_{z}(z \in x \land z\cup{z} = y)])$, considering that some axioms' description could be even bigger. Which is I thought it had more to say. – Schilive May 29 '22 at 00:15
  • What version of the axiom of infinity have you read? – Arvid Samuelsson May 29 '22 at 10:15
  • @ArvidSamuelsson, the same in Wikipedia. Sorry for taking so long to answer you. – Schilive May 29 '22 at 21:55

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In ZFC with the axiom of infinity excluded you can build sets of arbitrary size (you can make do with the axioms of the empty set and pairing), however the cardinality of the resulting set will always be finite. The axiom of infinity essentially asserts that there is a set that is one single object that is infinite in size. And yes, this means the smallest kind of infinity, "on top of which" other infinities (e.g. the continuum) are "built".

This is a very short answer, there is still a lot to say about the axiom, both from mathematical and philosophical points of view.

Alex
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  • Am I correct to say «The axiom of infinity asserts the existence of the kind of infinity of the set of the natural numbers, consequently asserting the existence of the set of the natural numbers, since any set with that kind of infinity contains everything that contains the set of the natural numbers»? Would this be a better interpretation? ALSO, thank you very much for taking the time to answer me. – Schilive May 29 '22 at 00:37
  • I 'm having a little trouble parsing what you're trying to say. Here's another point of view: the axiom of infinity is sometimes said to assert the existence of an inductive set (a set such that the empty set belongs to it, and that if some set belongs it, then its successor set does). The naturals are then defined to be the smallest inductive set, the existence of which can be proved. – Alex May 29 '22 at 00:43
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    Alex, you've kinda said in a comment what I said in a paragraph :). Thank you. I am happy that I think I understand the axiom now. Thank thee again. – Schilive May 29 '22 at 00:56
  • @Schilive you're welcome! – Alex May 29 '22 at 00:57