FOM: Friedman's independence results, an epochal f.o.m. advance wtait at
Fri Mar 13 22:49:54 EST 1998

Torkel Franzen wrote (3/13} in response to Steve's recommendation of 
Harvey's "FOM: 12:Finite trees/large cardinals"

> This may well be the case, but remains to be established, at least
>if f.o.m. is to be a subject of interest to non-specialists.
>  My own tendency as I attempt to penetrate the combinatorial
>principles here at issue is to lapse into slack-jawed wonder that
>anybody can make sense of them, let alone formulate them.
>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.

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. 
>there are probably very few who have anything approaching
>Friedman's peculiar ability to formulate, analyze and wrap his brain
>around principles of this type. 

What has this got to do with anything?

>  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. 

Have you checked web on ineffable cardinals? I've always wondered about 
that. (There probably have been lots of inaccessible cardinals.)
>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.

That can be done along the lines of the Goedel program of obtaining 
cardinals by closing under operations on ordinals that have already been 
introduced. In fact, one can obtain the stronger (and I hope more 
familiar) partition principles \kappa \arrow (stationary)^n_2, for each 
n, in this way. 

But you are missing the point of Friedman's program, which is precisely 
to justify the introduction of cardinals  by their low-down 
(combinatorial or whatever) consequences. One might want to argue that 
this is not a real justification; but an argunment would be needed.

Bill Tait

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