[FOM] Replacement
Jan Pax
pax0 at seznam.cz
Sat Aug 18 17:56:52 EDT 2007
> > Exactly, I have the same thing to point here: can the existence of
> > Hartogs function proved without Replacement?
> >
If we have a well ordered set W then h(W) is an ordinal whose initial segment is isomorphic to W.
By my previous post this requires replacement.
>
> > Another thing:
> > Consider the statement: If F: On --> P(S) be an increasing function,
> > where P(S) is the power set of non-empty S; then f(a) = f(a+) for some
> > ordinal a.
> > Can the above proved without Replacement? Or is there a counterexample
> > in ZF - Replacement ?
>
Take union of { F(alpha) \ alpha < | P(S) | }.
Some element from the preimage of this union and its successor will be the required a and a+.
Replacement is not needed.
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