Consequences of Vopěnka’s Principle over weak set theories

Athanassios Tzouvaras¹

¹ Department of Mathematics Aristotle University of Thessaloniki 541 24 Thessaloniki, Greece

Fundamenta Mathematicae, Tome 235 (2016) no. 2, pp. 127-152

Résumé

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.

Détail

DOI : 10.4064/fm982-1-2016

Keywords: shown vop nka principle restore almost entire weak fragment namely est theory consisting axioms extensionality empty set pairing union cartesian product delta separation induction along omega est proves axioms infinity replacement separation powerset result motivated previous friedmans where distinction made among various forms corollary est hbox foundation est hbox foundation zfc shown foundation axiom independent hbox foundation whether axiom choice independent weak form choice follows other similar forms choice introduced