Consequences of Vopěnka’s Principle over weak set theories

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, ∆0-Separation and Induction along ω, then 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, EST + Foundation + VP = ZF + VP and EST + Foundation + AC + VP = ZFC + VP. Also, it is shown that the Foundation axiom is independent of ZF − {Foundation} + VP. It is open whether the Axiom of Choice is independent of ZF + VP. A very weak form of choice follows from VP, and some other similar forms of choice are introduced.