First order Peano axioms and their intepretation

Question

I am still better trying to understand the first order Peano Axioms and their relation to the standard model. Just for reference, here are the axioms I work with:

$1.\space \forall x (\neg x+1=0)\hspace{2.14cm} 4.\space\forall x\forall y(x+1=y+1 \rightarrow x=y)\\ 2.\space\forall x(x+0=x) \hspace{2.41cm} 5.\space\forall x\forall y(x+(y+1)=(x+y)+1)\\ 3. \space \forall x(x\times0=0)\hspace{2.41cm} 6. \space \forall x\forall y (x\times(y+1) = (x\times y)+x)\\ $

As well as the axiom schema of infinite, but countable axioms for induction:

$$7. \space\forall x_1...\forall x_n(\varphi[0/y]\land \forall y( \varphi\rightarrow \varphi[y+1/y]))\rightarrow \forall y\varphi)$$

Now, for correctness to hold, whatever is deduced from these axioms, must hold in every interpretation. But, what I find weird is, it seems these axioms have already defined the function symbols and constants to be equivalent to the structure of $\mathfrak N$, ie. $\{+,\times,0,1\}$, instead of having undefined $f_1,f_2,c_1,c_2$, so as I understand it the mappings are already assigned. How is this allowed, since clearly that's just one interpretation of the functions/constants, while the deduction should apply to every interpretation?

Further if that's the case, do the different possible models of PA concern themselves only with the domain $A$ of the model? I read that there are non-standard models of PA for every cardinality, so what would a model with cardinality of one look like?

Any examples/sources for FO Peano deductions are of course also welcome.

Answer 1

But, what I find weird is, it seems these axioms have already defined the function symbols and constants to be equivalent to the structure of $\mathfrak N$, ie. $\{+,\times,0,1\}$, instead of having undefined $f_1,f_2,c_1,c_2$, so as I understand it the mappings are already assigned.

No, they haven't. The symbols $+$, $\times$, $0$, and $1$ used in the axioms are just symbols. There is no assumption that they refer to the usual meaning of addition, multiplication, zero, and one: $+$ could be interpreted as any binary operation, $\times$ as any binary operation, and $0$ and $1$ as any two constants. If you like, you could rewrite the axioms using $f_1,f_2,c_1,c_2$ as you suggest and the axioms would have the exact same meaning: you're just using different symbols.

I read that there are non-standard models of PA for every cardinality, so what would a model with cardinality of one look like?

This statement is not quite correct. There are non-standard models of PA of every infinite cardinality, but there are no finite models of PA. For instance, you can prove from PA that the elements $0$, $1$, $1+1$, $(1+1)+1$, $((1+1)+1)+1$, and so on are all distinct, so there are infinitely many different elements.

Answer 2

I thought The Incompleteness Phenomenon by Goldstern and Judah captured this well, but there must be other books on the subject as well. You have the Peano axioms and the standard model. A standard construction is to add a countable set of new axioms $c \gt 0, c \gt 1, c \gt 2, \ldots$ Any finite set of these axioms is satisfied by the standard model because you can find a greatest constant that $c$ must be greater than and choose it higher. Compactness now tells you that all the axioms have a model, but the standard model does not satisfy it, so there must be a nonstandard model. They state that all countable nonstandard models are of the form $\Bbb N$ followed by a dense linear order of sets isomorphic to $\Bbb Z$