FOM: re: CH / Funny subsets of RxR
Stephen Fenner
fenner at cs.sc.edu
Thu Sep 17 13:20:13 EDT 1998
Cohn's and Schlottmann's analyses of Riis's "proof" of not-CH are
well-taken and correct; there is no proof of not-CH here. However, Riis
it seems has rediscovered one of those pathalogical sets whose existence
owes both to CH and AC, namely, a (nonmeasurable) subset Z of [0,1]x[0,1]
such that:
forall x, { y | (x,y) in Z } is co-countable (hence measure 1)
forall y, { x | (x,y) in Z } is countable (hence measure 0)
In Riis's game, Player I chooses x and Player II essentially chooses y.
There are lots of other subsets of Euclidean space having weird,
counterintuitive measure properies if we assume AC and CH, and many of
them seem to depend on the fact that R has only continuum many open/closed
subsets.
Stephen Fenner 803-777-2596
Computer Science Department fenner at cs.sc.edu
University of South Carolina http://www.cs.sc.edu/~fenner
Columbia, SC 29208 USA
More information about the FOM
mailing list