[FOM] Eliminability of AC

Robert M. Solovay solovay at Math.Berkeley.EDU
Sun Mar 30 05:28:00 EDT 2008

I believe the original proofs of Ax and Kochen of their results 
concedrning p-adic fields used the theory of ultraproducts in a way that 
required GCH. Since the results obtained were arithmetic, one could then 
invoke the Kreisel remark to disentangle their results from GCH.

I don't recall at this late date whether Ax and Kochen made this 
observation themselves.

 	--Bob Solovay

On Sat, 29 Mar 2008, John Baldwin wrote:

> JOe asked for examples of theorems proved using absoluteless and then
> later the absoluteness was removed.
> ONe example is the proof that the independence property for n-ary formulas
> is equivalent to the independence property for 1-ary formulas. Shelah
> proved this by absoluteness and Laskowski gave a direct ZFC proof.
> I think there was similar example by Keisler in the 60's (eliminating gch)
> but I don't recall the exact statement. Perhaps this will jog someone's
> mind.
> On Mon, 24 Mar 2008, Thomas Forster wrote:
>> Does Shelah's work on the Whitehead problem count as a solution in
>> anyone's book...?
>> On Mon, 24 Mar 2008, joeshipman at aol.com wrote:
>>> One month ago, I noted the standard result that any use of the Axiom of
>>> Choice could be eliminated from proofs of arithmetical statements, and
>>> indeed from proofs of Sigma^1_2 statements, and asked the question:
>>> What is the simplest example of a well-known open problem  in "ordinary
>>> mathematics" (that is, one of interest to mathematicians in general and
>>> not primarily of interest to logicians and set theorists) where there
>>> is a possibility some form of Choice is needed for any proof?
>>> No one was able to provide one that met all three criteria (well-known
>>> AND open AND outside of logic and set theory), so I conclude that
>>> mathematicians outside of logic and set theory do not care about the
>>> Axiom of Choice anymore -- they are only interested in questions that
>>> are sufficiently absolute that their truth value does not depend on AC.
>>> It is possible that this increased emphasis on concrete problems
>>> compared to several decades ago is a reaction to forcing and the
>>> independence proofs, combined with the failure to isolate sufficiently
>>> plausible or useful new axioms.
>>> Since AC is an axiom one may use without explicit mention and still
>>> have a publishable paper, I don't see any remarks in current
>>> mathematical literature outside of logic and set theory that proofs do
>>> or do not depend on AC. This is more evidence that mathematicians do
>>> not care about AC.
>>> But Shoenfield Absoluteness goes further: not only may AC be eliminated
>>> from the proofs of arithmetical or Sigma^1_2 statements; so may V=L, a
>>> much stronger axiom. Can anyone provide examples, particularly
>>> arithmetical ones, of theorems outside of logic and set theory which
>>> were first proven (or are most easily proven) by showing they follow
>>> from V=L and then applying Absoluteness?
>>> -- Joe Shipman
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