[FOM] Schroeder-Bernstein dual.

Robert M. Solovay solovay at Math.Berkeley.EDU
Wed May 30 19:53:31 EDT 2007

Take a model where aleph_1 does not inject into the reals. Let X = the 
reals; let Y be the disjoint union of aleph_1 and the reals.

For example, take my model of "All sets Lebesgue measurable". Any variant 
of this example has at least the consistency strength of ZFC + " there is 
an inaccessible cardinal". I don't know offhand if the inaccessible is 
needed for a counterexample to your dual SB thm.

--Bob Solovay

On Tue, 29 May 2007, Bill Taylor wrote:

> Consider this "dual" to Shroeder-Bernstein:
> **  If there are surjections   f: X --> Y
> **                       and   g: Y --> X
> **
> **  then there is a bijection between X and Y.
> It follows easily from AC, but seems to be strictly weaker.
> Is there an easy model of ZF where this dual is false?
> Does it have any interesting equivalents?
> wfct
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