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?
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Collection is not provable in ZFC minus extensionality, a simple countermodel is described in http://mathoverflow.net/questions/54328 . (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 $n\in\bar\omega\smallsetminus\{\bar0\}$.) |
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