Notes on the equiconsistency of ZFC without the Power Set axiom and second order PA

Abstract

We demonstrate that theories ${\text{Z}}^{-}$, ${\text{ZF}}^{-}$, ${\text{ZFC}}^{-}$ (minus means the absence of the Power Set axiom) and ${\text{PA}}_2$, ${\text{PA}}_2^{-}$ (minus means the absence of the Countable Choice schema) are equiconsistent to each other. The methods used include the interpretation of a power-less set theory in ${\text{PA}}_2^{-}$ via well-founded trees, as well as the Gödel constructibility in the said power-less set theory.

Hopefully close to be the final version