[FOM] Dual of Dedekind-infiniteness.

[Vaughan Pratt]

I don't know much about forcing, certainly not enough to know how 
fine-grained it can be.  So what goes wrong when you try to force choice 
to hold for all families A_i whose elements, over all A_i, can be put in 
bijection with the indices, which is enough to show D* --> D, and then 
try to force choice to fail on some family where every A_i is bigger 
than 2^I say.  This would show that D* --> D is weaker than choice.

Vaughan Pratt

Lawrence Stout wrote:
> I looked at the variants on Dedekind finiteness in two papers:
> "Dedekind Finiteness in Topoi", JPAA 49 (1987), 219-225
> "Finiteness Notions in Fuzzy Sets", Fuzzy Sets and Systems, Vol.124  
> No. ER1 (2001?)
> 
> Neither of these is directly answering your question since they lie  
> outside the narrower setting of ZF or ZFC.  I remember that both  
> Sierpinski and Tarski had a lot to say about these.  Jech's book on  
> the axiom of choice gives some of the variants.  Troelstra  also has  
> a paper giving a large number of different intuitionistic variants on  
> finite.