FOM: MK and Introspection

[Harvey Friedman]

Reply to Shoenfield Tue, 22 Feb 2000 10:59.

>     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.

Its importance is that it apparently formalizes all of the obvious natural
axioms in its language, just as the system Z_2 does in its language. This
should be a theorem, although we don't know quite how to formulate it. In
other words, we need to explicate the notion of "obvious natural axiom" in
this and many other contexts, and this is certainly one of the great
problems of f.o.m. (A number of issues need to be sorted out such as how to
incorporate the versions of choice that can be formulated in this language.
One key point must be that choice does not raise the interpretation
degree.) The importance of MK is independent of the fact that we don't yet
have good results in this direction.

>     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 cited Simpson's book since Simpson uses Z_2 as important background
material to orient the reader. And look at the title of Simpson's book -
subsystems of second order arithemtic. I.e., the title is even "subsystems
of Z_2."

>     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 don't regard MK as a simplification of NBG. On the other hand, I think
that the difficulties involved with MK over NBG are generally worth the
added trouble.

>    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.

Very few mathematicians spend time thinking critically about what the truly
significant issues and programs are, at the level of continually being open
to rethinking the validity and appropriateness of their main lines of
research, taking into account the crucial issue of g.i.i. (general
intellectual interest). With most lines of research,  very substantial
limitations on what can be accomplished become apparent over time -
especially when g.i.i. is taken into account - and major rethinking of
their merits normally suggests major modifications. This kind of critical
rethinking - where cherished assumptions of importance and relevance that
looked convincing years ago now look dubious, and where g.i.i. is taken
property into acocunt - is normally absent in the thinking of most even
very good mathematicians. This is where the power of philosophical
introspection comes in.

>You give 14 examples
>of important results which supposedly make use of philosiphical
>introspection, but give no example of the philosophical intro-
>spection involved.

A common denominator in the relevant philosophical introspection involves
the realization that the issues being addressed are transcendentally more
important than virtually all other issues being addressed by others at the
same time period.

>     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.

The real problem is: what axioms should we adopt for normal mathematics?
That is the issue that is most interesting. In particular, the number of
set theorists is infinitesmially small compared to the number of normal
mathematicians, and it is now generally perceived among normal
mathematicians that set theory plays only a very marginal role in normal
mathematics - in the sense that only a very marginal portion of set theory
plays any role in normal mathematics.

In this sense, Steel's communications bear very little on the question
"what axioms should be adopted for normal mathematics". It is generally
perceived as obvious to the mathematics community that large cardinals are
completely irrelevant to normal mathematics.

They are being proved wrong, but not by the set theorists.

>In
>statibg that I thought Steel's comments said more about this problem
>then all previous fom communications, I did mean all.

Steel's comments said virtually nothing about the question "what axioms
should be adopted for normal mathematics", whereas dozens of my postings
say quite a lot about the question "what axioms should be adopted for
normal mathematics."

>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.

There are virtually no results proved by set theorists which are relevant
to the question "what axioms should be adopted for normal mathematics."
Furthermore, this question is not currently even on the agenda of the set
theory community.

>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.

These are not relevnat to "what axioms should be adopted for normal
mathematics." More relevant - but still not very relevant - than the
results you mention above are

1) Solovay's result that an uncountable coanalytic set has a perfect subset
from a measurable cardinal.
2) Results of Martin, Harrington, Steel, in connection with any two
analytic sets which are not Borel are Borel isomorphic.

Both of these results have the drawback that normal mathematicians have the
rather attractive alternative of:

*clarifying the notion of set they view themselves as ultimately working
with to that of constructible set*

thereby avoiding the impact of all of these and other independence results
proved by set theorists.

I am planning a detailed posting "What does it take?" in which I
systematically address the question of what kind of results are needed to
get normal mathematicians to take set theory seriously.