[FOM] Dual of Dedekind-infiniteness.

Bill Taylor W.Taylor at math.canterbury.ac.nz
Sat Oct 6 00:16:12 EDT 2007

Does anyone know if the following "dual" of Dedekind-infinitude
has been studied?

First let me just check the official definition of the basic idea:

D:  A set is D-infinite if it has a non-surjective injection into itself.

   Is that correct (to within logical equivalence)?   Then the dual is:

D*: A set is D*-infinite if it has a non-injective surjection onto itself.

It is simple that D implies D*, but the reverse doesn't seem to hold in ZF.

Does anyone know about this?  As I recall, Sierpinsky has some results
of this type, (e.g. he looks at the *Hartog's function - the surjective
non-equivalent version), but not this.  Similarly, I have been informed,
one cannot prove the surjective form of Schroeder-Bernstein in ZF,
(surjections both ways implies bijection), but it is strictly weaker than AC.

Bill Taylor

More information about the FOM mailing list