[FOM] Eliminability of AC
Francois G. Dorais
dorais at math.cornell.edu
Tue Apr 1 12:19:15 EDT 2008
James Hirschorn wrote:
> joeshipman at aol.com wrote:
>>> 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?
>
> On Wednesday 26 March 2008 20:34, Francois G. Dorais replied:
>> I'm not aware of any speedup associated with Shoenfield absoluteness.
>> If there is, I would really like to know.
>
> A basic example for absoluteness (but not exactly Schoenfield's absoluteness)
> is:
>
> Theorem. Every Delta-1-2 set of reals is Lebesgue measurable.
>
> Corollary. All analytic sets (i.e. Souslin sets, A-sets, Sigma-1-1) are
> measurable.
Yes. Another neat example is the original proof (using MA) of the
Baumgartner-Hajnal Partition Theorem vs Galvin's later combinatorial proof.
I believe there may be a general speedup theorem here, but I am not
aware of any general result of this kind. Again, I would really like to
know more about this (especially for Shoenfield absoluteness).
>> In any case, I'm pretty sure that you won't find any known arithmetical
>> theorem whose "easy proof" requires, e.g., using a morass and Shoenfield
>> absoluteness.
>
> I suspect you are mistaken, although I don't have an arithmetical
> counterexample off hand. For the example given in this thread on p-adic
> fields, perhaps the original proof is also the "easy proof"?
These are CH examples, if I'm not mistaken, which is much simpler than
V=L as a hypothesis. Formally, any proof of an arithmetical statement
that makes significant use of V=L has the initial hurdle of decoding the
statement of V=L. So there is a constant lower bound on the length of a
V=L proof. It is definitely possible that such an example exists, but I
suspect that human knowledge has not leaped that far yet...
I still claim that there is no known arithmetical statement whose "easy
proof" uses a morass (or some other significant chunk of V=L). An
example of that would be very interesting to see.
--
François G. Dorais
Department of Mathematics
Cornell University
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