Prove that the set of all integers $>0$ is the smallest inductive set

Question

An inductive set is a set $I$ such that $1 \in I$ and if $x \in I$ then $x+1 \in I.$

Some authors define the set of all integers $>0$ as the smallest inductive set, say Apostol's Analysis. But I am wondering, how to prove that the set of all integers $>0$ is indeed the smallest inductive set under, say the Peano axioms?

Answer 1

Perhaps the most useful way out of this problem is to say it cannot be done.

For, on the one hand, we have a construction of $\Bbb N_{>0}$ given some ambient structure (usually $\Bbb R$ as a complete ordered field) where we have defined $1$ and $+$.

On the other, we have a completely axiomatic description of what $\Bbb N$ is (the distinction between $\Bbb N$ and $\Bbb N_{>0}$ is not really material).

Because the definition of "inductive set" is particular to the former situation, we cannot expect it to carry over to the second because it refers to some ambient properties that we cannot describe using Peano's Axioms.

We can, of course, verify that the construction satisfies the axioms (this is actually an important step in establishing the usefulness of this construction of $\Bbb N$ in $\Bbb R$).

We could also go from $\Bbb N$ and build up $\Bbb Z, \Bbb Q, \Bbb R$, define $1$ and $+$ on this $\Bbb R$ (after having proved it is a complete ordered field), and then reconsider the problem, but this seems to me like a pointless circular exercise.

Answer 2

Let $X$ be a Dedekind-infinite set. Then, by definition, there exists $f:X\to X$ such that $f$ is injective but not surjective.

Since $f$ is not surjective, there exists $x_0 \in X$ such that $\forall y\in X:f(y)\ne x_0$. It is then possible to extract a subset $I\subset X$ such that $(I,f,x_0)$ satisfies the Peano Axioms where $f$ is used as the successor function, and $x_0$ as the zero (first element). $I$ will be the smallest subset of $X$ such that $x_0\in I$, and if $x\in I$ then $f(x)\in I$.

Note, too, that all ordered triples $(N,S,0)$ that satisfy the Peano Axioms are order-isomorphic, i.e. they are essentially identical in structure and size. In some sense then (you will have to tweak your definition a bit), the set of natural numbers is the smallest inductive set.