Re: Separation,Power and Countability.


On Jun 19, 7:26 am, zuhair <zaljo...@xxxxxxxxx> wrote:
|I asked for the proof of the following:
|For any set d that is a member of Pw and is indefinable what is the
|prove of d being equinumerous to w?

Meaning, I suppose, how can one prove that each
undefinable subset A of w is equinumerous with w.

If A is finite, then A is definable as {n : n is
a natural number and (n=a0 or n=a1 or ... or
n=a_m)} for some a0,...,a_m.

If A is undefinable, then, A is infinite and each
infinite subset of w is equinumerous with w. Let
a0 be the least element of A, and inductively let
a_{k+1} be the least element of A not among a1,...,
a_k. Then a_i gives a 1-1 correspondence between
A and w.

Keith Ramsay

.