[FOM] Dual of Dedekind-infiniteness.
Vaughan Pratt
pratt at cs.stanford.edu
Mon Oct 8 14:46:38 EDT 2007
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.
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