FOM: re: CH / Funny subsets of RxR

Stephen Fenner fenner at
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

Stephen Fenner                       803-777-2596
Computer Science Department          fenner at
University of South Carolina
Columbia, SC  29208  USA

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