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I know that the set of all natural numbers is defined as an infinite set under the Axiom of Infinity, but I'm having a hard time using $\exists S [\emptyset \in S \land (\forall x\in S)[x \cup {x} \in S]]$ to define an infinite set composed of all real numbers, for all integers, or just for all real numbers between $0$ and $1$. An explanation would be greatly appreciated.

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  • $\begingroup$ In set theory, real numbers are sets. $\endgroup$ – Asaf Karagila Aug 30 '13 at 5:29
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There are many good questions here about constructions of the real numbers. For example Completion of rational numbers via Cauchy sequences.

Note that the Axiom of Infinity is only used to guarantee existence of the ordinal $\omega$. The rest of the ordinals and other sets follow from the other axioms.

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    $\begingroup$ That could be misleading. Without the Axiom of Infinity, you don't get any infinite ordinals. What you mean is that the Axiom of Infinity gives you $\omega$, while you need the Axiom of Infinity plus the other axioms to get you the rest of the ordinals. $\endgroup$ – Peter Smith Aug 30 '13 at 5:31
  • $\begingroup$ Related. $\endgroup$ – Andrés E. Caicedo Aug 30 '13 at 5:39

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