[FOM] Status of AC
Harvey Friedman
friedman at math.ohio-state.edu
Thu Mar 2 01:38:34 EST 2006
On 3/1/06 9:39 PM, "joeshipman at aol.com" <joeshipman at aol.com> wrote:
> 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.
Every Pi^1_4 sentence provable in ZFC is provable in ZF.
If my memory is correct, many years ago an example of a Sigma^1_4 sentence
provable in ZFC but not in ZF was given.
I think it goes something like this.
1. There is a model of ZF with infinitely many c-degrees, but no infinite
sequence of c-degress. c-degrees are the degrees of constructibility of
sugbsets of omega.
2. ZFC refutes this.
3. In particular, ZFC proves the negation of the following:
i. For all n there exists n distinct c-degrees; and
ii. There is no infinite sequence of distinct c-degrees.
Note that i is of the form
(forall n)(therexists x)(forall y)(therexists z).
Note that ii is Pi^1_3.
3. So the statement that ZFC proves the negation of is (somewhat better
than) Pi^1_4. Hence ZFC proves a Sigma^1_4 sentence that ZF doesn't.
Again if my memory is correct, in the old days people had given an example
of a
(therexists n)(forall x)(therexists y)(forall z)
statement provable in ZFC but not in ZF.
Harvey Friedman
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