Toronto Set Theory Seminar
Friday, April 22 from 1:30 to 3pm
Fields Institute, Room 210
Speaker: Joel Hamkins (CUNY)
Title: What is the theory ZFC-Powerset?
Abstract: The theory ZFC-, consisting of the usual axioms of ZFC but with the Powerset axiom removed, when axiomatized by Extensionality, Foundation, Pairing, Union, Infinity, Separation, Replacement and the Axiom of Choice, is weaker than commonly supposed, and suffices to prove neither that a countable union of countable sets is countable, nor that $omega_1$ is regular, nor that the Los theorem holds for ultrapowers, even for well-founded ultrapowers on a measurable cardinal, nor that the Gaifman theorem holds, that is, that every $Sigma_1$-elementary cofinal embedding $j:Mto N$ between models of the theory is fully elementary, nor that $Sigma_n$ sets are closed under bounded quantification. Nevertheless, these deficits of ZFC- are completely repaired by strengthening it to the theory obtained by using Collection rather than Replacement in the axiomatization above. These results extend prior work of Zarach. This is joint work with Victoria Gitman and Thomas Johnstone.