Amendment to Re: FOM: Friedman's independence results, an epochal f.o.m. advance
wtait at ix.netcom.com
Sat Mar 14 09:37:31 EST 1998
An apology (in both senses). In the posting in question (yesterday,
Friday the 13th), in response to Torkel Franzen, I wrote
>Certainly many people would agree with Steve (and Harvey) that a proof
>that _any_ combinatorial problem can be solved only using a particular
>large cardinal axiom is of foundational interest.
which is silly, since with natural assumptions, a large cardinal axiom
always implies a new consistency statement. I intended my remark to be
directed at Torkel's
>>What is needed to convince people
>>that these are "very natural combinatorial propositions" is to find
>>some striking applications of them. Even I could no doubt grasp
>>these principles if I set my mind to it, but I need some incentive.
And I meant to say that, for _any_ combinatorial problem, a proof that it
can be solved only using a particular large cardinal axiom is of
foundational interest. (In elementary logic classes they teach students
about proper placement of quantifiers.)
What may be at issue here, though, is Torkel's qualification
>if f.o.m. is to be a subject of interest to non-specialists.
I tend not to take the `general intellectual interest' condition on fom
very seriously (to the extent that I even understand it). One of the
outstanding foundational problems (unless one takes to heart Sol
Feferman's argument from need) is certainly that of giving meaning, if
possible, to large cardinal statements. It is difficult to see how one
could understand this foundational problem without going to the effort to
understand the LC statements involved.
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