## Consequences of Vopěnka’s Principle over weak set theories

### Volume 235 / 2016

#### Abstract

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, $\Delta _0$-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 ones (2014), as well as by H. Friedman's (2015), where a distinction is made among various forms of VP. As a corollary, ${\rm EST}+ \hbox {Foundation} + {\rm VP}={\rm ZF+VP}$ and ${\rm EST}+ \hbox {Foundation} + {\rm AC+VP}={\rm ZFC+VP}$. Also, it is shown that the Foundation axiom is independent of ${\rm ZF} - \{\hbox {Foundation}\} + {\rm VP}$. It is open whether the Axiom of Choice is independent of ${\rm ZF+VP}$. A very weak form of choice follows from VP, and some other similar forms of choice are introduced.