[FOM] Role of Polemics
Harvey Friedman
friedman at math.ohio-state.edu
Sun Jan 22 23:55:06 EST 2006
On 1/21/06 8:09 PM, "Neil Tennant" <neilt at mercutio.cohums.ohio-state.edu>
wrote:
> In the kind of polemics you describe, does the person arguing on your side
> of the debate have to agree to use only certain kinds of methods (to
> establish relevant independence results, for example)? Or can that person
> use any kind of method ever used in mathematics, for constructing or
> defining mathematical objects, and for reasoning about them?
>
In my posting http://www.cs.nyu.edu/pipermail/fom/2006-January/009598.html I
made an attempt to be unusually general in the discussion of a certain kind
of f.o.m. polemic.
I suggest that if we want to take the discussion into more specific channels
such as the above, that we be more specific. E.g., you could outline a
specific polemic of this kind, and we could discuss the kind of response I
would be seeking to it.
Perhaps a useful way to think of a main point I was making is this.
I believe that the following holds for any branch B of mathematics. For
every seriously proposed set theoretic axiom A to date, there is an
explicitly Pi01 sentence phi such that
a) it is provable in EFA (exponential function arithmetic) that Con(ZFC + A)
iff phi;
b) phi is a statement that many practitioners in B would find sufficiently
interesting to discuss in a Seminar in their branch - in particular,
sketching proofs of close variants of phi that are provable well within ZFC;
c) various changes of numerical parameters in phi will create various Pi01
sentences that are equivalent (provably in EFA) to the consistency of a
representative sample of standard set theories ranging from finite set
theory through the strongest large cardinal axioms commonly considered.
I hold related opinions about what I call "completely finite" statements.
These are statements where there is an absolute upper bound on the number of
objects under consideration. Even with absolute upper bounds like 8!!.
Harvey Friedman
More information about the FOM
mailing list