FOM: MK and Introspection

Martin Davis martin at
Tue Feb 22 17:52:49 EST 2000

At 10:59 AM 2/22/00 -0500, Joseph Shoenfield wrote:

>      Finally, let me clarify my remarks about Steel's communication.
>I took the phrase "problems of large cardinals" from your communi-
>cation; they do indeed include your a), b), and c).   But I really
>consider all this to be part of a larger problem, which, roughly
>stated, is: what axioms should we adopt for set theory?   Large
>caridnals are part of the solution, not part of the question.   In
>statibg that I thought Steel's comments said more about this problem
>then all previous fom communications, I did mean all.   But I did
>not mean to imply that I have gone through each such comment and
>compared it to Steel's.   I meant that no comment I have seen on
>this question takes into account the many results proved by set
>theorists which might be relevant to the problem.   I believe that
>some of the fairly recent results, such as the Martin-Steel theorem
>and the results of Wooden and Steel quoted in Steel's communication
>are very relevant.

I certainly agree with Joseph Shoenfield about the importance and 
significance of this work, which, to my mind, is complementary to Harvey 
Friedman's important success in transforming into finite 
combinatorial  statements every mathematician can appreciate at sight, 
metamathematical consequences of the existence of large cardinals 
demonstrably transcending ZFC.


                           Martin Davis
                    Visiting Scholar UC Berkeley
                      Professor Emeritus, NYU
                          martin at
                          (Add 1 and get 0)

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