[FOM] Eliminability of AC

[John Baldwin]

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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John T. Baldwin
Director, Office of Mathematics and Computer Education
Department of Mathematics, Statistics, and Computer Science
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