# Is the set of all real numbers defined under the Axiom of Infinity in ZFC?

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.

• In set theory, real numbers are sets. – Asaf Karagila Aug 30 '13 at 5:29

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.
• 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. – Peter Smith Aug 30 '13 at 5:31