[FOM] Cylindric Algebra and Consistency
Vaughan Pratt
pratt at cs.stanford.edu
Mon May 30 02:25:15 EDT 2011
On May 28, 2011, Dana Scott wrote:
> Perhaps there is a misunderstanding here? Is it not well known that
> every arithmetic formula over the ring of integers can be put into
> prenex normal form with just one (mammoth) polynomial equation as the
> matrix?
Ah, sorry, terminology snafu. I took your "polynomial variety" too
literally, when what you meant (obviously in retrospect) was
"diophantine set." I misunderstood you to be trying to make number
theorists and algebraic geometers feel at home (diophantine sets and
polynomial varieties over Q intermingle in algebraic number theory).
I hadn't seen (or had forgotten) that nice trick for making the PA
matrix diophantine and hence r.e. Without it the matrix could denote a
more obscure set, though how much does this matter when the subsequent
complementing and projecting is going to scramble things at least as
obscurely?
I agree that cylindrification was a great way to make logic more
appealing to mathematicians. Which came first, that or Halmos's
polyadic algebras? (I encountered the latter while an undergraduate
browsing the Sydney Uni. library stacks in 1965, I first encountered
cylindric algebras more than a decade later.)
Vaughan
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