FOM: Re: V=L vs. PD

[John Steel]

On Sat, 13 May 2000, Harvey Friedman wrote:

> Reply to Steel Fri, 12 May 2000 14:55.
> 
>The normal mathematician is deliberately completely uninterested in
> foundational issues of the sort we are talking about, because they think
> that it is only relevant for mathematics concerned with full
> generality....
>> The normal mathematician is aware that there are axioms for mathematics
> that he is constantly using, but does not recall what they are, nor ever
> had a really good understanding of them in the first place.

         This is one reason I object to placing such great weight on this   
person's opinions.
     The other reason is that "normal" is much too vague, not to
mention value-laden. You are changing the question to "Does normal
mathematics need new axioms?". We need a clearer idea of what counts
as normal in order to discuss that.
    We could take our question to be "Does finitary combinatorics need   
axioms beyond ZFC?". My short answer to that question would be: Godel's   
incompleteness theorems say this might be the case, the applications of 
large cardinals in the realm of countable sets make it more likely, and
recent results of Friedman may have brought this abstract possibility to
fruition.
     
 
> >Obviously theories need applications.
> 
> The matter of applications is far far more critical than this sentence
> suggests in the case of set theory. There are special difficulties with set
> theory that are completely uncharacteristic of any special difficulties
> encountered in normal mathematics. And these special difficulties are
> inexorably intertwined with a nearly universal feeling of suspicion - at
> various levels - of set theory. There is a general feeling that, as a
> matter of principle, such abstract objects, apparently completely remote
> from any arithmetical, algebraic, or geometrical considerations, must be
> inherently useless for any arithmetical, algebraic, or geometrical
> purposes, including any connections with science and engineering.
    The
> objects of higher set theory are naturally regarded as unnatural foreign
> intruders into normal mathematics.
> 
> So it is not merely a matter of set theory needing applications. It is a
> matter of a deep suspicion that set theory cannot possibly have any
> applications - as a matter of principle - except to mathematics (such as
> set theory) that is concerned with full generality.
> 
            The belief that large cardinals cannot, as a MATTER OF
PRINCIPLE, have applications to finitary combinatorics, was refuted
decisively by Godel in 1931. Once again, our "normal" mathematician just
seems to be ill-informed.
            In your list "arithmetical, algebraic, or geometrical...",
you leave out "analytical". Intentionally? Lebesgue did not think that
the question of Lebesgue measurability of projective sets is remote from
Analysis.

 
> Secondly, I am not saying that it would have been rational to reject large
> cardinal theory, or Zermelo set theory, or ZFC, until something like
> Boolean relation theory is done.
> 
     Of course, we agree here. I think this is a significant point.



> What I am criticizing is the lack of identification as to what the really
> crucial issue for large cardinals - or set theory for that matter - has
> been. The set theory community has consistently acted like this was not the
> major crucial issue.

      There's room for more than one crucial issue. If THE crucial issue
to which you are referring is whether there are applications of higher set
theory in finitary combinatorics, it's not clear that set theorists are
the right people to be working on this. If you are referring to
applications in a broader sense, then it's just not true that set
theorists do not seek these.

 Instead the idea was promulgated that the issue of
> just what good set theory is for anything has already been fully resolved
> for all significant parties by determinacy, starting at the Borel level.
> 
      I don't know any set theorists who would subscribe to that idea.


> >Harvey's own work would never have been done in that case.
> >Mahlo cardinals
> >would never have been discovered by looking through the lens of Boolean
> >Relation Theory.
> 
> The first sentence here is false or at best very misleading. The hierarchy
> of Mahlo cardinals that figures so prominently in Boolean relation theory
> and earlier work of mine, was invented by Mahlo in a series of papers in
> 1911, 1912, 1913. Steel can hardly be referring to prohibiing or
> discouraging this work of Mahlo in the years 1911, 1912, 1913. At that
> early time, there was hardly anything around to prohibit or discourage. And
> let me just say that if Mahlo did this in 1911 - 1913, then any of a number
> of people - including me - could have been expected to do this by the year
> 2000. And, of course, only basic elementary work is needed for Boolean
> relation theory in connection with Mahlo cardinals. I certainly have known
> about Schmerl's Ph.D. thesis in the 1970's with Jack Silver about the
> Ramsey theoretic aspects of the Mahlo cardinal hierarchy. But again, the
> relevant part of his work lies within the range of many people after the
> Mahlo hierarchy is set up and one wants to do the combinatorics of Mahlo
> cardinals.
> 
      I think both sentences are true, and they contain an important
point. Setting up the Mahlo hierarchy, and investigating its
Ramsey-theoretic aspects, are very pure set theory. It's hard to imagine
a finite combinatorist doing this because he saw the need in some
problem of finite combinatorics.
     As I understand it, Harvey himself started with the pure set theory.
Whether he could have invented himself is beside the point--the point
is that the intellectually natural progression in this case is
theory-to-application.
    Even if some brilliant finite combinatorist had started with questions
in Boolean relation theory, and then somehow made his way to Mahlo
cardinals, he would have had to ask "Are there Mahlo cardinals?". (Of
course, after he had invented metamathematics and proved Godel's theorems,
he might have realized that 1-con(Mahlos) was all he needed to know.) This
would have led him to develop the ideas around Mahlo cardinals; i.e. to
become a set theorist. (By the way, if you change "combinatorist" to
"analyst", "questions in Boolean relation theory" to "questions in
trigonometric series", and "Mahlo" to "uncountable", you get something
like Cantor's story.)


> 
> >Large cardinal hypotheses flow from general foundational
> >considerations,
> 
> Absolutely true, but even here I have a sharp disagreement with the set
> theory community. They have not recognized the need for, and the realistic
> possibilities of, obtaining far better intrinsic justifications for even
> small large cardinals than we have today. In fact, we need, and ought to
> look for, deeper intrinsic justifications even at the levels of ZFC and
> Zermelo. Flow is not enough.
> 

     Are you dismissing Reinhardt's work?


John Steel