FOM: Re: Axioms for normal mathematicians

[Harvey Friedman]

Reply to Shipman Mon, 06 Mar 2000 15:00.

>>These are not relevant to "what axioms should be adopted for normal
>>mathematics." More relevant - but still not very relevant - than the
>>results you mention above are
>>1) Solovay's result that an uncountable coanalytic set has a perfect
>subset
>>from a measurable cardinal.
>
>What is the precise result?  Do you need a measurable and not just its
>consistency?  In the other direction, are large cardinals or their
>consistency necessary?

Solovay proved that over ZFC, "uncountable coanalytic sets have perfect
subsets" is equivalent to "there are countably many reals constructible
from any given real." The latter is provable in ZFC plus "there exists a
measurable cardinal". Also he proved that the systems ZFC + "uncountable
coanalytic sets have prefect subsets" and
ZFC + "there exists a strongly inaccessible cardinal" are equiconsistent.
In particular, ZFC + "uncountable coanalytic sets have perfect subsets"
proves the consistency of ZFC, and in fact the consistency of ZC + "there
exists an inaccessible cardinal."

Also, in particular, ZFC + V = L proves "there is an uncountable coanalytic
set without a perfect subset."

>>2) Results of Martin, Harrington, Steel, in connection with any two
>>analytic sets which are not Borel are Borel isomorphic.
>
>What axioms beyond ZFC were used to show this and what axioms have been
>shown necessary?

ZFC proves that "any two analytic sets of reals which are not Borel can be
mapped onto each other by a Borel bijection of the reals" is equivalent to
"for all reals x, x# exists." In particular,  "any two analytic sets of
reals which are not Borel can be mapped onto each other by a Borel
bijection of the reals" is provable in ZFC + "there exists a measurable
cardinal." Also, e.g., ZFC + "any two analytic sets of reals which are not
Borel can be mapped onto each other by a Borel bijection of the reals"
proves the consistency of ZFC + "there exists a cardinal arrowing omega"
and the consistency of ZC + "for all reals x, x# exists."

Also, in particular, ZFC + V = L proves "there exists two analytic sets of
reals which are not Borel and which cannot be mapped onto each other by a
Borel bijection of the reals."

>>Both of these results have the drawback that normal mathematicians have
>the
>>rather attractive alternative of:
>>
>>*clarifying the notion of set they view themselves as ultimately
>working
>>with to that of constructible set*
>>
>>thereby avoiding the impact of all of these and other independence
>results
>>proved by set theorists.
>
>If "normal mathematicians" must concern themselves with "constructible
>sets" in Godel's sense this is already a big concession to set
>theorists;

Yes and no. If anything, it is a concession to Godel. But this can be done
in such a way that the mathematician would not be concerned with
constructible sets in Godel's sense in their everyday activities. They
would simply be told once and for all that a variety of statements hold in
the constructible universe, whatever that is, and that they can merely
assume these statements. E.g., there exists a PCA well ordering of the
reals (which I proved long ago was equivalent to all reals are
constsructible in some real), the continuum hypothesis, diamond, etcetera.
If they ever run into set theoretic problems, they can then pull these
statements out.

Of course, the modern concept of normal mathematics is well within the
demonstrably absolute, so the descriptive set theoretic conundrums don't
appear in normal mathematics.