From: Fred Galvin Subject: Re: Strategy vs Tactics. was: Famous Quote Date: Tue, 1 May 2001 12:56:27 -0500 Newsgroups: sci.math Summary: cardinalities/orders of partitions of R On 1 May 2001, David Libert wrote: > A difficult case which we can't rule out happening over ZF is the > reals partioning into sets A, B each of cardinality strictly less > than c = 2^Aleph_0. > > I will give a ZF proof ruling out a trivial special case of this, > which will be useful below. > > Though we presently don't see how to refute such A,B in ZF, ZF > does prove the special case: > > 2-partioning Claim: there are no A,B partitioning the reals > with A countable and |B| < c. I think the easiest way to see that is by using the fact that there's a bijection between R and RxR. So suppose RxR is partitioned into two subsets A and B. If A is countable, then A meets only countably many horizontal lines, so B must contain a horizontal line, so B has cardinality c. In the same way, it's easy to see that, if R is partitioned into two sets A and B, and if A is well-orderable, then at least one of A and B has cardinality c.