Consequences of Vopěnka's Principle over weak set theories

Abstract: It is shown that Vop\v{e}nka's Principle (VP) can restore almost the entire ZF over a weak fragment of it. Namely, if EST is the theory consisting of the axioms of Extensionality, Empty Set, Pairing, Union, Cartesian Product, $\Delta_0$-Separation and Induction along $\omega$, then ${\rm EST+VP}$ proves the axioms of Infinity, Replacement (thus also Separation) and Powerset. The result was motivated by previous results in \cite{Tz14}, as well as by H. Friedman's \cite{Fr05}, where a distinction is made among various forms of VP. As a corollary, ${\rm EST}+$Foundation$+{\rm VP}$=${\rm ZF+VP}$, and ${\rm EST}+$Foundation$+{\rm AC+VP}={\rm ZFC+VP}$. Also it is shown that the Foundation axiom is independent from ZF--{Foundation}+${\rm VP}$. It is open whether the Axiom of Choice is independent from ${\rm ZF+VP}$. A very weak form of choice follows from VP and some similar other forms of choice are introduced.