[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>

> 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

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