TITLE: Axiom to exclude nonstandard natural numbers
QUESTION [7 upvotes]: In Peano Arithmetic, the induction axiom states that there is no proper subset of the natural numbers that contains 0 and is closed under the successor function. This is intended to rule out the possibility of extra natural numbers beyond the familiar ones. It doesn't accomplish that goal; there remains the possibility that other natural numbers exist and the familiar ones do not form a set. In Internal Set Theory (IST), which is an extension of ZFC that is consistent relative to ZFC, there is a distinction between standard and nonstandard sets, and it can be shown that
(1) 0 is standard;
(2) if $n \in \mathbb{N}$ is standard, then so is its successor;
(3) $\mathbb{N}$ has nonstandard elements.
The induction axiom is not violated because the standard natural numbers do not form a set.
Is there a way to axiomatize set theory so that no such nonstandard natural numbers can exist?
(Note: this question is not about nonstandard models of arithmetic. In IST, $\mathbb{N}$ is a standard set, and within a given model of IST, all models of second-order Peano arithmetic are isomorphic to $\mathbb{N}$.)

REPLY [2 votes]: By Gödel's incompleteness theorem, there can't be any such axiom in a first-order, recursively enumerable theory.
You can axiomatize $\mathbb N$ by adding an infinitary rule of inference, the Hilbert $\omega$-rule:
$$P(0)\wedge P(1) \wedge P(2) \wedge \cdots \over \forall n P(n)$$
to PA for each arithmetic predicate P.  This says, if predicate P holds for each of the natural numbers (0,1,2...) then you can deduce the formula $\forall n P(n)$.  The resulting system is called  ω-logic.  
Obviously this is something of a "cheat" since you no longer have an effective theory.  As one example (maybe there are better ones) of how it can be used, Michael Rathjen's article "The Art of Ordinal Analysis" describes using the $\omega$-rule to analyze stronger and stronger arithmetic theories, and is pretty interesting.