Consequences of Vopěnka's Principle over weak set theories

It is shown that Vopěnka’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, ${\mathrm{\Delta }}_0$-Separation and Induction along $\omega$, then $\mathrm{E}\mathrm{S}\mathrm{T}+\mathrm{V}\mathrm{P}$ 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, $\mathrm{E}\mathrm{S}\mathrm{T}+$Foundation$+\mathrm{V}\mathrm{P}$=$\mathrm{Z}\mathrm{F}+\mathrm{V}\mathrm{P}$, and $\mathrm{E}\mathrm{S}\mathrm{T}+$Foundation$+\mathrm{A}\mathrm{C}+\mathrm{V}\mathrm{P}=\mathrm{Z}\mathrm{F}\mathrm{C}+\mathrm{V}\mathrm{P}$. Also it is shown that the Foundation axiom is independent from ZF–{Foundation}+$\mathrm{V}\mathrm{P}$. It is open whether the Axiom of Choice is independent from $\mathrm{Z}\mathrm{F}+\mathrm{V}\mathrm{P}$. A very weak form of choice follows from VP and some similar other forms of choice are introduced.