FOM: MK and Introspection

Joseph Shoenfield jrs at
Tue Feb 22 10:59:25 EST 2000

     This is a reply to Friedman's reply to me on 15 Feb 2000.
     Let me make some brief remarks on MK.   You say that MK is natural
and important.   I have already agreed that it is natural from some
points of view.   If importance is to mean something more than natural,
it must mean that we have proved non-trivial results about MK which
give us a better understanding of some concepts which we already be-
lieved are important.   I don't think anything like this has happened,
although it certainly might in the future.
     You have tried to show the importance of MK by analogy with the
system Z_2 of second order arithmetic, citing Simpson's book.   I have
not seen the book, but my understanding is that it devoted to certain
subsystems of Z_2 which are (in some sense) much weaker.   The fact
that they are all subsytems of Z_2 is not important, and no general
resuls about Z_2 are used in a significant way.  Perhaps you feel
I am wrong about this.
     I would like to make clear that my original unfavorable remarks
about MK were not an implied criticism of your use of MK in the
result you proved.   They were a criticism of those who have into-
duced MK as a simplification of NBG without realizing the many added
difficulties the were introducing.
    I think we have much more important differences on the use of
"philosophical introspection".    (I am omitting the adjective "deep",
which seems to serve no purpose except to make philosophical intro-
spection seem more important.)
    You say "the most common use of philosophical introspection in
fom is the realization of what the most significant issues and
programs are".   It seems to me that all good mathematicians in
every field spend time deciding what the significant issues
and programs are.   Is all of this philosophical introspection?
If not, what characterizes the part which is?   You give 14 examples
of important results which supposedly make use of philosiphical
introspection, but give no example of the philosophical intro-
spection involved.
     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.

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