SOME PRELIMINARIES: Predicate logic is consistent and complete. In other words, (i) for a closed formula $F$ in predicate calculus with equality and functions, $\vdash F$ if and only if $\,\vDash F$ (where $\vDash F$ means $F$ is true under the standard interpretation of the logical constants for any assignment of predicates and functions occurring in $F$). Furthermore, (ii) if $\,\vdash F$ in first-order arithmetic, then for some finite sequence of formulas $\Gamma$ (where $\Gamma$ are closed axioms of Peano arithmetic), $\Gamma \vdash F$ in predicate calculus with equality and functions.
Now here is my argument, where did I make a mistake. Suppose $\vdash F$ in first-order arithmetic. Then by (ii), $\Gamma \vdash F$ in predicate logic. Thus $\vdash \Gamma' \rightarrow F$ (where the formula $\Gamma'$ is the conjunction of the formulas in $\Gamma$). By (i), $\vDash \Gamma' \rightarrow F$. Then, in the standard model of arithmetic (and all other models), $\Gamma' \rightarrow F$ is a true statement under the interpretation. And in intuitive number theory, the proposition $\Gamma'$ is true in the standard model. Thus intuitively, $F$ must be true. Therefore, if $F$ is provable in first-oder arithmetic then it is true intuitively. Then if first-order arithmetic were inconsistent, the proportions $0=0$ and $0\neq0$ would be provable and thus both true in the standard model, which is absurd. Therefore the formal system must be consistent.
Is this even a valid argument? Is this a strong argument or is a more of a heuristic argument because it appeals to non-finitary methods? Is it circular because it relies on intuitive number theory being consistent? Furthermore, if this argument isn't valid, why do we formalize number theory if we cannot know that the theorems are necessarily true?
Best Answer
The fact that $\mathbb{N}$ is a model of first-order Peano arithmetic (hereafter simply Peano arithmetic) is sufficient to show the consistency of Peano arithmetic.
However, the fact that we are able to talk about the set $\mathbb{N}$ and how it "models" arithmetic assumes that we are working in a set theory (such as ZF) as our meta-theory (or at least some stronger meta-theory than Peano arithmetic). Thus, we are proving the consistency of Peano arithmetic in an even stronger theory than Peano arithmetic (which itself could be inconsistent).
We cannot prove the consistency of Peano arithmetic within Peano arithmetic itself (unless Peano arithmetic is actually inconsistent, in which case we have much bigger problems). This follows from Gödel's incompleteness theorem.
So essentially, the fact that Peano arithmetic is consistent can be taken for granted philosophically as much as any statement proved in ZF can be.