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Oct 29, 2009 at 0:08 comment added Richard Dore Start with L and add omega2 many Cohen reals. In the extension, the reals of L is a definable set of reals with intermediated cardinality.
Oct 18, 2009 at 17:53 comment added Richard Dore One can definitely prove aleph(P^3(S)) > aleph(S). The proof is a good hard exercise in basic set theory (it's in chapter 1 of Kunen). It doesn't require knowing anything sophisticated, just running the diagonalization argument carefully. I don't know offhand what better bounds can be achieved than that.
Oct 18, 2009 at 8:06 history answered Kenny Easwaran CC BY-SA 2.5