How do Peano's arithmetical axioms guarantee that we can construct the natural number set?

Question

I'm probably not understanding but I don’t see how the axioms can guarantee total construction of the natural number set.

The successor function as well as the axiom of induction guarantee that the natural number set will not just infinitely loop over including the same element, and that it will grow to infinite cardinality.

But isn’t it possible that it can go

{0, 1, 5, 6, 7, 8, 9, ...} thereby leaving “holes” in the set?

Again, probably a misguided view of sets, but I feel like this can only be solved by doing a union between all the (infinitely many) generated sets from all the different trials/worlds, in order to completely populate the ultimate natural number set?

Or my other thinking is that the successor function must be defined for all the different algebraic presentations of all the natural numbers, and then the sets produced by it must undergo a union.

But then how does the successor function even equate "next" with +1. It's like using a word in its own definition, and then abstracting it to "next" in order to obfuscate the cyclical nature of the definition. It even feels like the axioms mix up Sets and Series.

Please help me smooth over this confusion.

Answer 1

The set of natural numbers from Peano is $$\{0,\,S(0),\,S(S(0)),\ldots\}\ .$$ As an abbreviation, you can call these anything you like. But if you want to use the symbol $5$ to abbreviate $S(S(0))$ and $6$ for $S(S(S(0)))$ and so on, things are going to get confusing very rapidly.

"Successor" does not mean adding $1$. Addition is defined by the axioms $$x+0=x\ ,\quad x+S(y)=S(x+y)$$ for all $x,y$. It is not true by definition that $S(x)=x+1$, but it is a theorem which you can prove from the axioms, together with the definition $1=S(0)$.

Answer 2

$0$, $S(0)$, $S(S(0))$ are terms that we can indeed interpret in any way we want ... and indeed the function symbols $s$, $+$, and $\times$ can likewise be interpreted in many different ways.

For example, I could say that $0$ denotes 'apples', $S(0)$ denotes 'bananas', $S(S(0))$ denotes 'Luxemburg', etc.

So you're right that the domain is not at all forced to be the natural numbers and their typical operators of successor, addition, and multiplication.

However, given that from the Peano Axioms we can prove things like:

$\forall x \ x \times 0 = 0$

$\forall x \ x \times S(0) = x$

$\forall x \ x \times S(S(0)) = x + x$

etc.

you will find that 'apples' behaves just like $0$, that 'bananas' behaves just like $1$, and that 'Luxemburg' behaves just like $2$. That is, any model of the eano Axioms will have to behave just like the natural numbers: we say that the models are isomporph. So ... you might as well treat $0$ as $0$, $S(0)$ as $1$, etc.