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, ${\Delta }_0$-Separation and Induction along $\omega$, then $\mathrm{EST}+\mathrm{VP}$ 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, $\mathrm{EST}+$Foundation$+\mathrm{VP}$=$\mathrm{ZF}+\mathrm{VP}$, and $\mathrm{EST}+$Foundation$+\mathrm{AC}+\mathrm{VP}=\mathrm{ZFC}+\mathrm{VP}$. Also it is shown that the Foundation axiom is independent from ZF--\{Foundation\}+$\mathrm{VP}$. It is open whether the Axiom of Choice is independent from $\mathrm{ZF}+\mathrm{VP}$. A very weak form of choice follows from VP and some similar other forms of choice are introduced.

22 pages

References in corpus (1)

• Large transitive models in local {\rm ZFC}