[FOM] Status of AC
joeshipman@aol.com
joeshipman at aol.com
Wed Mar 1 21:39:18 EST 2006
Nguyen:
>I have a question: what is the current status of AC among
mathematicians and
>philosophers of mathematics?
For practically all the most important mathematical questions, the use
of AC is eliminable, because of metatheorems like Shoenfield's
Absoluteness Theorem, which ensures that AC can be eliminated from the
proof of any statement of logical type no higher than Sigma^1_2 or
Pi^1_2. (In fact, not only AC but allso V=L is eliminable and can
therefore can be used freely when attempting to prove such statements)
The famous "open questions" of mathematics are generally of much lower
type than that -- the Riemann hypothesis, Poincare conjecture, and "P
not = NP" are all equivalent to arithmetical statements of low logical
complexity (Pi^0_1, Pi^0_1, Pi^0_2 respectively). These are three of
the seven Clay Institute "Millennium Prize" problems, and three others
are also officially formulated by the Clay institute in ways that can
be shown to be "absolute". (The seventh, on Yang-Mills gauge theories,
is not formulated clearly enough that I can tell whether or not it is
absolute).
Even a much more abstract statement like the Invariant Subspace
Conjecture ("every bounded linear operator on Hilbert space has a
nontrivial invariant subspace") is Absolute because it can be shown
equivalent to a Pi^1_2 statement by coding elements of l2 as real
numbers.
Can anyone identify a well-defined mathematical question that is
currently considered by mainstream mathematicians to be both
"important" and "open", to which the Shoenfield Absoluteness Theorem
does not appear to be applicable?
-- Joseph Shipman
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