[FOM] Schroeder-Bernstein dual.

Robert M. Solovay solovay at Math.Berkeley.EDU
Wed May 30 20:33:37 EDT 2007

Upon further reflection:

Start with a model M of V=l. Force to add a countable sequence of Cohen 
generic reals a_{i,j} indexed by pairs of integers.
Let b_i = {a_{i,j}: j in omega}.
Let c = {b_i: i in omega}.
Let N be the submodel of M[<a_{i,j}: i, j in omega>] consisting of all 
sets which are OD from finitely many of the a's, b's, and c.

I say that in N the dual of SB fails. For this it suffices to show that c 
does not inject into the reals; but this is an easy symmetry argument.

 	--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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