[FOM] Historical Queries on AC

Alasdair Urquhart urquhart at cs.toronto.edu
Wed Jan 9 18:32:14 EST 2008

On Tue, 8 Jan 2008, joeshipman at 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).
> Was this published before or after Godel's 1938 PNAS paper proving the
> consistency of GCH and AC?
> If Tarski's paper came first, there is a sense in which he, not Godel,
> was the first mathematician to provide a consistency proof of AC
> acceptable to modern mathematicians (because the Tarski "Universes
> axiom" is freely used by modern mathematicians in algebraic geometry
> and other core mathematical areas).

Tarski's paper does not seem to have a date of reception, but at the
end of the paper, Tarski says (my translation from the German):

"The author presented a report on the results contained in this
article on 18/VI/1937 to the Polish Mathematical Society."

Goedel's extended abstract of 1938 in the Proceedings of the National
Academy of Sciences, U.S.A., is headed "Communicated November 9, 1938"
(this is missing from the reprint in the Collected Works, Volume II).

However, Goedel had already presented his early work on the constructible
sets in a summer course "Axiomatik der Mengenlehre", starting
in May 1937, and in that course, he showed that AC holds in the universe
of constructible sets.  There exist shorthand notes by Goedel for the
course, as well as reminiscences of Mostowski, who attended
the course (in Crossley's  "Reminiscences of Logicians") -- see the Dawson 
biography, p. 122.

Hence, if we count the Goedel lecture course and the Tarski lecture
as publications (as I think we should), the priority still seems
to belong to Goedel.

Alasdair Urquhart

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