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.
Categories: math.LO