Collection from Replacement in ZFC-extensionality - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T23:55:03Z http://mathoverflow.net/feeds/question/116837 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/116837/collection-from-replacement-in-zfc-extensionality Collection from Replacement in ZFC-extensionality Frode Bjørdal 2012-12-20T02:40:05Z 2012-12-20T13:59:14Z <p>My question is whether the axiom of extensionality is required to show that the schema of collection follows from the schema of replacement in the usual Zermelo-Fraenkel environment with choice. In other words: Is the schema of collection a theorem schema in Zermelo-Fraenkel set theory with choice minus the axiom schema of extensionality?</p> http://mathoverflow.net/questions/116837/collection-from-replacement-in-zfc-extensionality/116859#116859 Answer by Emil Jeřábek for Collection from Replacement in ZFC-extensionality Emil Jeřábek 2012-12-20T12:19:25Z 2012-12-20T13:59:14Z <p>Collection is not provable in ZFC minus extensionality, a simple countermodel is described in <a href="http://mathoverflow.net/questions/54328" rel="nofollow">http://mathoverflow.net/questions/54328</a> . (That the model cannot provably satisfy collection follows from Gödel’s theorem. For a specific instance of collection which fails, let $\bar\omega$ denote one of the many representations of $\omega$ in the model, and $\bar0\in\bar\omega$ the corresponding empty set: then the model satisfies “for every $n\in\bar\omega\smallsetminus\{\bar0\}$, there exists a function $f$ with domain $n$ such that $f(\bar0)=\bar\omega$, and $f(x)\in f(y)$ whenever $x\in y\in n$”, but there is no set collecting such functions for every $n\in\bar\omega\smallsetminus\{\bar0\}$.)</p>