# FOM: Martin-Steel theorem

Stephen G Simpson simpson at math.psu.edu
Wed Sep 9 13:18:12 EDT 1998

```Joseph Shoenfield writes:

> By a strictly pi-0-n+1 sentence I mean one which is not pi-0-n or
> sigma-0-n.

I still don't understand.  By adding dummy quantifiers, it is trivial
to convert a Pi^0_1 sentence to a logically equivalent Pi^0_{n+1}
sentence which is not Pi^0_n or Sigma^0_n.

You could avoid this difficulty by talking about sentences up to
equivalence over ZFC or something of the sort, but I still don't see
the relevance of this.

> I stated this conjecture only because you demanded one,
> but I would really rather return to my original general statement:

OK, let's drop the discussion of your conjecture, which makes no sense
to me anyway.

> one should look for a result which relates the position of an
> undecidable statement in the arithmetical or analytic hierarchy and
> the number and kind of large cardinals needed to prove it.

Why?  I don't see that such a relationship would have a bearing on any
important f.o.m. issue.  It seems *much* more important to find good
examples of finite combinatorial statements that require large
cardinals to prove them.

> I have always been puzzled as to why you considered the particular
> result of Harvey such a key result in the completeness program.

The reason I view Harvey's independence result as key is that it is
the state of the art vis a vis the program that I mentioned, i.e. to
extend the incompleteness phenomenon into finite combinatorics, or
more specifically, to find finite combinatorial statements which are
independent of ZFC, or ZFC plus large cardinals.  By state of the art
I mean the best result that is known at the present time.

> I find your challenge to explain why I value the Steel-Martin
> theorem so highly not only reasonable but welcome, since it will
> afford me a chance to express some thoughts on judging mathematical
> results which I have had recently.  I hope to meet the challenge
> sometime soon.

I'm looking forward to it.  I wonder how you are going to express your
thoughts on these matters without using any informal concepts.

-- Steve

```