Jesse Alama wrote:
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?
It is in Tarski's paper:
Alfred Tarski
On Well-ordered Subsets of any Set,
Fundamenta Mathematicae, vol.32 (1939), pp.176-183
Actually his goal was to prove that the existence of sufficiently large
cardinals is enough to get Axiom of Choice, look to
the title of 1938 paper:
Alfred Tarski
Ueber unerreichbare Kardinalzahlen,
Fundamenta Mathematicae, vol.30 (1938), pp.68-89
When Tarski was accused that he can prove the Axiom of Choice only because of
the specific form of Axiom A
(roughly speaking the existence of universal classes) then he had introduced
the Axioms B (existence of arbitrary large strongly inaccessible cardinals)
and proved the _equivalence_ of both axioms (A and B).
I am in Nagano now, and my access to the literature is a bit restricted,
could anybody look to 1939 paper?
However, I believe that the proof of the correctness of 'Rank' (CLASSES1:def
6) does not depend on TARSKI:9 (or ZFMISC_1:136, if you like), or it should
not, and then the theorem
theorem :: CARD_LAR:37
M is strongly_inaccessible implies Rank M is being_Tarski-Class;
does the trick. (Still, we have to look to references in the proof of
CARD_LAR:37 - particularly to the references to the theorems in CLASSES1 - if
they are independent of TARSKI:9).
Josef, what you think, it is ypur article: CARD_LAR, so you know better.