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 $\mathbb{N}$.)