# Has anyone successfully axiomatized the category of finite sets? In such a way that the resulting theory is bi-interpretable with PA.

In studies of ZFC, it is conventional to take Peano arithmetic (hereafter PA) as the metatheory. However, I don't like this convention; I think a better approach would be a metatheory (like ZF-fin with $\in$-induction, or whatnot) that describes $V_\omega$, the universe of hereditarily finite sets. This creates a cleaner interface between the metatheory and the object theory, in my opinion.

Now suppose we instead wish to study ETCS. We could take Peano arithmetic as our metatheory; but, by the above argument, this probably isn't the right thing to do. We're better off having a metatheory that describes the category of finite sets. So I ask: Has anyone successfully axiomatized the category of finite sets in such a way that the resulting theory is bi-interpretable with PA?

• This may be helpful: R. Diaconescu and L.A.S. Kirby, Models of arithmetic and categories with finiteness conditions, Annals of Pure and Applied Logic, Volume 35, 1987, Pages 123-148 – Lawrence Wong Jan 10 '14 at 9:54

I don't know if this fits to your question, but the category of finite sets is the free finitely cocomplete category on one object. That is, if $C$ is a category with finite coproducts and coequalizers, and $X \in C$, there is essentially a unique functor $F : \mathsf{FinSet} \to C$ preserving finite coequalizers and coproducts such that $F(\star)=X$, where $\star \in \mathsf{FinSet}$ is the terminal object. (For a characterization of $\mathsf{Set}$, replace "finitely cocomplete" by "cocomplete"). See here for another characterization involving the cartesian monoidal structure.