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[mizar] [Thomas Forster] Re: [FOM] Historical Queries on AC

Judging from Forster's reaction and questions that I've received from
subscribers to the FOM mailing list concerning my post on TG, it looks
like there's some surprise that Choice is a theorem of TG.  The surprise
is that the universe axiom implies AC.  That was my intuition as well;
those two don't seem to be related to each other.  Can anyone provide an
intuitive sketch of why that follows?

Thanks,

Jesse
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On Tue, 8 Jan 2008, joeshipman@aol.com wrote:

> 1) In 1938, Tarski (Fund. Math. vol. 30) showed that AC follows from
> the axiom that there is a Universe containing any set (in other words,
> that arbitrarily large inaccessible cardinals exist). Of course, the
> consistency (rather than the truth) of AC doesn't need the full
> Universes axiom, just one inaccessible limit of inaccessibles (because
> that set will satisfy ZF and the Universes axiom).


You mean that if there is a proper class of strong inaccessibles then AC 
follows..?   Surely you can't mean that!?!

> 3) Sierpinski's proof is is stronger than just "GCH-->AC", it actually
> shows that for a set A to be well-orderable one needs only that there
> are no intermediate cardinals anywhere in the sequence A < P(A) <
> P(P(A)) < P(P(P(A))) < P(P(P(P(A)))).  Has anyone improved this to
> require a smaller set of no-intermediate-cardinal assumptions?
>
> 4) Analogously to 3), what is the best known result on how many levels
> of Universes above A are necessary in order to well-order A?


Say $\alpha$ adj $\beta$ if there is no cardinal intermediate between the 
two.  Then alpha adj 2^alpha adj 2^2^alpha implies that 2^alpha is an 
aleph.  This last refinement is due to Ernst Specker i think. To do it he 
used a lemma that says that if x> 5 then x^2 < 2^x.  (The proof is quite 
hard: try it!)  As far as I know it is still an open question whether or 
not alpha adj 2^alpha implies that alpha is an aleph.  That sounds 
surprising, and i expect to be told that it was solved at some point when 
i wasn't paying attention...

   The person who knows all the detailed history of this is 
John Truss in Leeds.


  -- 
URL:  www.dpmms.cam.ac.uk/~tf; DPMMS ph:
+44-1223-337981. Mobile +44-7887-701-562.








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-- 
Jesse Alama (alama@stanford.edu)