[FOM] Status of AC
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:
>> 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
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
(therexists n)(forall x)(therexists y)(forall z)
statement provable in ZFC but not in ZF.
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