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The cardinality of the natural numbers is $\aleph_0$.

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In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite sets.

Q: Which axioms of ZFC are required to prove the existence of the cardinal number $\aleph _ 0$?


closed as unclear what you're asking by Tim Raczkowski, Asaf Karagila, user230715, Lord_Farin, user642796 Aug 30 '15 at 5:20

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    $\begingroup$ What do you mean "the cardinal number $\aleph_0$"? Do you mean a set of size $\aleph_0$, or do you mean the ordinal $\omega$, or do you mean perhaps the class of all countably infinite sets? $\endgroup$ – Asaf Karagila Aug 27 '15 at 16:56
  • $\begingroup$ I mean an object used to denote the size of the set of natural numbers? Is that not the usual meaning? $\endgroup$ – Dan Christensen Aug 27 '15 at 17:31
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    $\begingroup$ That is the set of natural numbers. $\endgroup$ – Asaf Karagila Aug 27 '15 at 17:33
  • $\begingroup$ What is confusing is whether you ask what is needed to "prove" the existence of a countably infinite set (there is an Axiom of Infinity) or you mean what is needed to prove the machinery around cardinal numbers that goes with ZFC's broader theory of sets. Asaf is asking (reasonably IMHO) what you mean by $\aleph_0$. Your question could be interpreted narrowly or broadly. $\endgroup$ – hardmath Aug 27 '15 at 18:47
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    $\begingroup$ Unless and until you can specify exactly what you mean by $\aleph_0$ there is no point in attempting to answer this question. Usually $\aleph_0$ is synonymous with $\omega$, the least infinite ordinal, or the set of all finite ordinals, or the set of all natural numbers. However you seem to be in a state of disbelief about this, so you must have in mind some other object denoted by $\aleph_0$. It is incumbent on you to describe this object sufficiently well enough that others can answer your question. $\endgroup$ – user642796 Aug 30 '15 at 5:23

I think all you need is:

  • The axiom of infinity
  • The schema of separation (or at least, an instance of it).

The idea is:

  • First, use infinity to get an inductive set $X$.
  • Now let $\omega$ equal $$\{x \in X \mid x \mbox{ belongs to every inductive set}\}.$$
  • $\begingroup$ You wrote: "Write $\mathcal{I}$ for the collection of all inductive subsets of $X$, which exists by separation." Separation on what set? $\endgroup$ – Dan Christensen Aug 27 '15 at 16:27
  • $\begingroup$ @DanChristensen, $\mathcal{I} = \{A \in \mathcal{P}(X) \mid A \mbox{ inductive}\}.$ $\endgroup$ – goblin Aug 27 '15 at 16:29
  • $\begingroup$ @DanChristensen, okay, here's a better way of doing it. $\endgroup$ – goblin Aug 27 '15 at 16:31
  • $\begingroup$ But $X$ is just one inductive set. $\endgroup$ – Dan Christensen Aug 27 '15 at 16:31
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    $\begingroup$ @Dan: You only need the one. If there are any other inductive sets, their intersection with X is a subset of X. $\endgroup$ – Malice Vidrine Aug 27 '15 at 16:38

Metamath's theorem omex proves it from the Axiom of Extensionality, the Axiom of Union, the Axiom of Separation, the Axiom of Pairing, the Null Set Axiom, and the Axiom of Infinity.

  • $\begingroup$ Woah that much? I wonder what's wrong with my proof? $\endgroup$ – goblin Aug 27 '15 at 16:43
  • $\begingroup$ @goblin: You can examine the proof to see the nitty-gritty details if you like, but they already need essentially all this machinery to prove the Peano axiom that any set containing zero and closed under the successor operation contains all natural numbers. $\endgroup$ – Charles Aug 27 '15 at 16:51
  • $\begingroup$ According to the commentary, that proves "The existence of omega (the class of natural numbers)." Is that really the same as proving the existence of the cardinal number $\aleph_0$? $\endgroup$ – Dan Christensen Aug 27 '15 at 16:51
  • $\begingroup$ @DanChristensen: Depends on your definition of aleph numbers, I suppose. You could replace omex above with aleph0 if you prefer. $\endgroup$ – Charles Aug 27 '15 at 16:53

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