Does mathematical induction assume that non-negative integers are infinite?

Does mathematical induction assume that the non-negative integers continue indefinitely? A friend of mine was attempting to show me that there are an infinite number of non-negative integers using mathematical induction but I always thought that, since one of the two main parts of mathematical induction relies on showing that what holds for $n$ must also hold for $n+1$, it already assumes that non-negative integers are infinite in number, for it assumes that there will always be a $n+1$ (that is, there will always be a successor). However, after seeing a formulation of the counting numbers using nested sets and mathematical induction then used to show that these constructions must be indefinite in number, I am not sure what to believe (I didn't understand the proof very well). Life doesn't make sense anymore. Please help me.

• Induction doesn't assume an infinity of naturals, but it assumes something (the existence of $n+1$) from which you can easily prove the infinity of naturals. – Gerry Myerson Jun 6 '13 at 10:09
• You can do induction on finite sets too (though it feels somehow far less interesting), as well as on sets that are not the naturals. – Billy Jun 6 '13 at 10:44
• The map $f\colon n\mapsto -n$ from the positive integers to the negative integers is bijective, because it's the inverse of itself. – egreg Jun 6 '13 at 11:56

One can have other structures, such as $\mathbb{Z}/5\mathbb{Z}$, that are finite but nevertheless have an $n+1$ for every $n$. Here the relevant difference from $\mathbb{N}$ is that zero does have the form $n+1$ for some $n$. This example shows that it is possible to have a structure with a total operation—which can therefore be iterated indefinitely—but such that this iteration does not produce an unlimited number of distinct elements. Just as for $\mathbb{N}$, the induction schema is valid for $\mathbb{Z}/5\mathbb{Z}$. However, such abstract considerations are unnecessary here because the structure is finite and one can simply write down the whole structure.
The point of the induction schema for $\mathbb{N}$ is that it allows us to get a handle on the structure even though we cannot simply write down the whole thing. In particular, it means that $\mathbb{N}$ is not "too large" and that no natural number itself is infinite. It is mainly the role of the other axioms to prove that $\mathbb{N}$ is not "too small". For example, without using induction one could prove that $\mathbb{N}$ has more than 5 elements, or more than 5,000,000 elements. (Although it is true that induction is required for a formal proof that $\mathbb{N}$ is infinite, because we have to prove that for all $n \in \mathbb{N}$ the structure $\mathbb{N}$ does not have size $n$, not just for $n=5$ or $n = \text{5,000,000}$, and induction is generally required to prove such universal statements.)