FOM: Friedman's independence results, an epochal f.o.m. advance

Martin Davis martind at
Fri Mar 13 20:40:47 EST 1998

>  Also, it is a significant circumstance that the only occurrence of
>the phrase "subtle cardinal" on any web page indexed by AltaVista is a
>reference to Cardinal Granvelle. To establish Friedman's results as
>important progress in f.o.m., a principle that yields the existence of
>subtle cardinals must be established as a comprehensible and
>potentially acceptable addition to the axioms of set theory.

As Harvey has explained to me, his condition is implied by the existence of
a measurable cardinal and even by the existence of 0 sharp. He went to some
trouble to find the weakest large cardinal axiom that would give the result.
(In the other direction, familiar large cardinal axioms that he can
demonstrate won't yield his results are the existence of strongly
inaccessible, Mahlo, or weakly compact cardinals.

As to the theorem being complicated, I've been digesting it, and sure it's
complicated. But, my God, it's an amazing breakthrough. I'm with Steve on
this one. (Maybe he'll stop fretting over trivialities about Boolean rings
vs. Boolean algebras.)


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