FOM: set-theory-1 and -2; Martin-Steel; mathematical incompleteness

[Stephen G Simpson]

Set-theory-2 and mathematical incompleteness:

  Joseph Shoenfield 16 Sep 1998 23:37:20 writes:
  
   > Steve and Harvey have replied to my posting on the Martin-Steel
   > theorem.  For both, the main point is that the problems of set
   > theory which I described are less important than certain other
   > problems.
  
  Huh?  I assume that Joe is referring to my 13 Sep 1998 13:18:55
  posting on set-theory-1 versus set-theory-2.  In that posting I was
  careful *not* to say that one is more important than the other.  I
  said that *the distinction* is important: set theory as a particular
  branch of mathematics (set-theory-1) versus set theory as a
  foundation for mathematics as a whole (set-theory-2).  I never said
  that either perspective is "less important".
  
  Judging from Joe's further comments, I think that Joe either doesn't
  fully recognize the above distinction or doesn't fully appreciate the
  set-theory-2 perspective.
  
   > .... This feature can also be described (in terms which both Steve
   > and Harvey have used) as: the problem has interactions with core
   > mathmatics.
  
  I didn't use exactly those terms.  I did discuss the mathematical
  incompleteness program, i.e. the program of extending the
  incompleteness phenomenon into various branches of mathematics,
  including core mathematics.
  
   > As the above shows, it is very diffucult to convince a person
   > that a criterion for the importance of a mathematical problem is
   > particularly significant if he does not alread believe this.  
  
  Yes, if that person is Joe Shoenfield.  :-)
  
   > If (as seems even more unlikely) Steve and Harvey convinced the
   > leading set theorists to spend their time working on such problems,
  
  I wasn't trying to talk the leading set theorists out of their
  perspective.  I was only pointing out the existence of a different
  perspective.
  
   > the result would be a disaster for set theory.
  
  This remark intrigues me.  Why do you think it would be a disaster?
  
  Shoenfield writes:
  
   > [My own comment is] an attack on a certain fom perspective, which
   > is not mine but may be close to Steve's.  It is that one should
   > first decide, using ones intuition, what the fundamental concepts
   > of the subject are and then concentrate on problems concerning
   > these.
  
  That's not my perspective exactly.  My perspective is as follows: I
  am moderately interested in set-theory-1; I am *very* interested in
  set-theory-2 and f.o.m. generally.
  
   > If the set theorists I mentioned had adopted this perspective, set
   > theory would have been a much less interesting subject than it is
   > today.
  
  I don't really see that.  What I would conjecture is that if the
  mathematical incompleteness program is successful, then set theory
  will be of much *greater* interest than it is today, and to a much
  wider audience.

More on Martin-Steel:

  Apparently Joe and perhaps others do not adequately appreciate the
  perspective of set-theory-2 and the mathematical incompleteness
  program.  Therefore, let me try to illustrate this perspective by
  using it to comment on Martin-Steel.
  
  The Martin-Steel theorem says that under appropriate large cardinal
  hypotheses we can generalize classical results about analytic and
  coanalytic sets to higher levels of the projective hierarchy.  This
  is a wonderful use of large cardinals.  However, from the
  perspective of mathematical incompleteness, moving *up* the
  projective hierarchy is the wrong direction.  For mathematical
  incompleteness, we must move down closer to the kinds of objects
  that occur in ordinary mathematical practice.  In terms of Harvey's
  "can't run away" posting, it's easy for mathematicians to run away
  from higher projective sets, but it's not so easy to run away from
  words on a finite alphabet.  Absoluteness represents a barrier to
  progress in this direction, but Harvey's recent work shows that the
  barrier may not be insurmoutable.
  
  The above remarks are made from what I regard as the set-theory-2 or
  mathematical incompleteness perspective.  It goes without saying
  that other perspectives on Martin-Steel are possible, and I respect
  and appreciate Joe's set-theory-1 perspective.  Set-theory-1 is a
  perfectly respectable branch of mathematics.

Historical questions:
  
  Let me close by asking some historical questions.  Harvey in 9 Sep
  1998 11:03:06 and 14 Sep 1998 03:09:43 says or implies that
  mathematical incompleteness was already an issue in the 1930's.  My
  question is, to what extent did G"odel and others in 1930's
  recognize the importance of the mathematial incompleteness program
  vis a vis G"odel's first and second incompleteness theorem?  To what
  extent did they recognize the existence of major obstacles to
  carrying out the program?  Can anyone supply quotes or references on
  this?

-- Steve