[FOM] Cantor Bendixson and the Axiom of Replacement
D.R. MacIver
drm39 at cam.ac.uk
Wed Jan 5 18:18:11 EST 2005
I do apologise - this ends up being much more trivial than I thought it was.
If you consider the map P(E) -> P(E) given by A -> { x : x is a limit point
of A } then A is perfect iff it's a fixed point of this map. But if you
consider P(X) ordered by reverse set inclusion (i.e. a <= b iff b \subseteq
a ), this ends up being a monotonic map on a complete poset, and by
tarski's fixed point theorem (which doesn't use the axiom of replacement
anywhere) has a smallest fixed point. i.e. there exists a largest perfect
set.
So it is indeed a theorem of Z set theory. Oh well.
David MacIver
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