FOM: large cardinals and Diophantine equations

[Martin Davis]

I've been surprised and gratified to see so much attention paid in this
forum to a result in which I played a part. I must admit that I much prefer
calling it MRDP (as the Europeans do) rather than "Matiyasevich's theorem"
and certainly not for any lack of admiration for Yuri M. His construction of
a Diophantine definition of the relation m=F(2n) where F(n) is the nth
Fibonacci number, is beautiful and succinct (and did something I'd been
trying to do for a decade).
But as I'm sure all of you know that construction only leads to the
conclusion that all r.e. sets are Diophantine on the basis of previous work
by Julia Robinson, Hilary Putnam and me.

But what lured me into changing from an observer to a participant was
Vaughan Pratt's assault on large cardinals. It seems to me that making sense
of them is THE most important and troublesome foundational problem of the
day. After all, by MRDP we know that these assumptions have consequences
asserting the non solvability of specific Diophantine equations. This
connection of the very large with very ordinary mathematics suggests that
(amending Frege) "the role of the [very large] infinite in arithmetic is not
to be denied".

Set theorists, having been given the power by the method of forcing to
create models of set theory at will, can easily think of themselves as
simply studying these structures as group theorists study groups. But of
course each of these countable structures contains a replica of the full
corpus of contemporary mathematics.

In the light of the above, I was particularly struck by Harvey Friedman's
offer to show how large cardinal axioms can appear as part of a natural
process of proceeding from the finite to the infinite. That sounds really
interesting.

Martin Davis