[FOM] Eliminability of AC
joeshipman at aol.com
Mon Mar 24 09:07:02 EDT 2008
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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