FOM: Does Mathematics Need New Axioms?

[John Steel]

Here is a reply to some of Simpson's posting.

On Thu, 18 May 2000, Stephen G Simpson wrote:

 
Simpson: What are the appropriate axioms for *all* of mathematics?
 
Steel:  Does this sound like a question which belongs in a "small and
        obscure corner"?
 
Simpson: From the standpoint of general intellectual interest, the
        question is  of tremendous importance.  But from the standpoint
         of the
         core or mainstream mathematician, the question belongs to a
        small and obscure corner.
        That corner is known as 03-XX, Logic and Foundations. 



      We seem to agree that set theory deals with an age-old question of
great importance. Steve wants to say that the question is no longer of
importance for "core" mathematics, although he concedes some mild
plausibility to the scenario that one day large cardinal hypotheses will
be widely applicable there.  As I said in the outline to my panel
statement: 

     "In order to avoid pointless wrangling about what is mathematics, I
would prefer to reformulate our question:

     Is the search for, and study of, new axioms worthwhile? Should people
be working in this direction?

This gets to the practical question."

     If Steve answers the question above negatively, it must be because he
rejects set theory as a framework for answering the age-old question. In 
this connection I would like to repeat a question from my previous posting 
which Steve did not answer in his last one:

Steve wrote: 

> There are alternative
> foundational schemes that many people find attractive.

I asked:
 
    Which schemes are you talking about? The various constructivist
schemes are fragments of ZFC. They are not incompatible with ZFC unless
one takes them as outlawing the mathematics which they cannot support.
Is this what you mean by an alternative scheme--the proposal to abolish
some part of ZFC or its large cardinal extensions?


John Steel