[FOM] Status of AC

[joeshipman@aol.com]

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