FOM: Axioms for normal mathematicians

Joe Shipman shipman at
Mon Mar 6 15:00:51 EST 2000


>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
>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?

>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?

>Both of these results have the drawback that normal mathematicians have
>rather attractive alternative of:
>*clarifying the notion of set they view themselves as ultimately
>with to that of constructible set*
>thereby avoiding the impact of all of these and other independence
>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; if you are attributing a different notion of "constructible"
to them, presumably one which doesn't include some analytic and
coanalytic sets, can you be more precise? Do you mean they will stay
within the Borel universe, or something in between Borel and analytic?

-- J. Shipman

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