Using Peano axioms to define natural numbers

Question

I am having some issues using the Paeno axioms to prove that closure under addition exists within the natural numbers. I think that a large part of my issue stems from my confusion over the notation used by my professor.

We are given that N is defined as:

N := { x $\in$ Z | $\exists$ m $\geq$ 1 s$^m$(0) = x}

I guess I am most confused by what "m" is. Does "m" is just a way of 'counting' the "s's" in the set? Am I overthinking this? Any help would be appreciated!

Answer 1

There is a problem with your professor's definition. We more or less already need the notion of the collection of natural numbers satisfying the Peano Axioms, even before it makes sense to talk about iterating a function, in this case $s$. In fact, the standard way to define iterates is:

Let $s^0$ be the identity function on $\mathbb{Z}$.

Let $s^{n+1} = s^n \circ s$ for every natural number $n$.

How can we possibly do this without the natural numbers? We essentially cannot, because the natural numbers are precisely what we need to use to count the iterations!

The reason I can say that your professor is making a mistake is that the quantification of $m$ is ill-defined; it quantifies over all "$m \ge 1$", but what is that even supposed to mean? It is in fact meaningless unless $m$ is restricted to some kind of number, not to say that "$s^m$" is meaningless unless $m$ is a natural number (or integer if $s$ is invertible; presumably it is not in this case). So it already is necessary to know $\mathbb{N}$ before any of the notation makes sense.

However, if you do things carefully in set theory, you could first construct an inductive set $ω$, and then somehow define the integers $\mathbb{Z}$ without using natural numbers, and then define the natural numbers via the recursion theorem. This is not going to be anywhere as 'simple' as the ill-defined definition quoted. For details of how to do this rigorously in ZFC, see this post. If a different formal system is being used, then he will have to specify it.

Answer 2

Usually the integers are constructed from (i.e. defined in terms of) the natural numbers, not the other way around. Usually the natural numbers are defined using some version of Peano's Axioms. Then the integers are defined in terms in the natural numbers, typically as follows::

$\forall a: [a\in Z \iff a\subset N^2 \land a\neq \emptyset$

$\land \forall b,c:[(b,c)\in a \implies \forall d,e\in N:[(d,e)\in a \iff b+e=c+d]]]$

Then addition on $Z$ is usually defined as a subset of $Z^3$. (I can't remember the exact form.)

Perhaps your professor meant his statement to be a theorem about $N$ and $Z$ rather than a definition of $N$. Even in that case, however, I think the theorem would be poorly stated. It is no wonder you are confused.

Answer 3

As you already have Z as the ordered ring of integers your definition just says N is the set of positive integers. You are asked to prove that the positive integers are closed under addition. In your definition $s$ is the successor function and the exponent $m$ is repeated application, so $m=x$. What you would like to say is given $x,y \in $N you can find $m,n$ such that $s^m(0)=x, s^n(0)=y,$ then $ x+y=s^{m+n}(0)$, so $x+y \in $N. As others have said, this is a strange order of definitions, and whether this is easy to prove will depend on what those definitions are.