Something has gone entirely wrong here. The axiom of regularity forbids x\in x. colin On Mon, May 28, 2012 at 11:15 PM, Fred E.J. Linton <fejlinton@usa.net> wrote:
On Mon, 28 May 2012 07:25:19 PM EDT, Florian Lengyel <florian.lengyel@gmail.com>, protesting my assertion that
Virtually no one ever wants to restrict attention to functions that respect (preserve or reflect) membership (other than "preserve" between ordinals).
remonstrated that
Within set theories that satisfy the axiom of regularity, one's attention is restricted to functions that both preserve and reflect self-membership.
f(x) \in f(x) iff x\in x
Hereto, I in turn ask: Why only self-membership? why not membership outright
-- f(x) \in f(y) if (and/or only if) x \in y -- ?
And how often, really, do we actually impose either of those restrictions? (Or
did FL inadvertently omit a "sometimes" between "is" and "restricted" :-) ?)
Cheers, -- Fred
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