[FOM] 303: PA Completeness (restatement)
A.P. Hazen
a.hazen at philosophy.unimelb.edu.au
Mon Oct 30 23:09:02 EST 2006
Harvey Friedman has made the interesting conjecture that PA is
complete for arithmètic sentences containing no more than two
quantifiers. If true, I think this would qualify as a SURPRISING
result: most of the classical work on PA has regarded "bounded"
quantifiers as free (which is equivalent to allowing the basic
non-logical vocabulary of the language to be enriched with symbols
for arbitrary primitive recursive functions and relations). And of
course PA is very far from being complete for sentences with two
UNBOUNDED quantifiers! (So, my guess is that many of us have
"intuitions" that are not properly tutored for thinking about
Harvey's conjecture. And that if it turns out to be true we will
therefore be surprised by it.)
On the more general topic of whether First-Orlder Logic (FOL) is the
"Appropriate" logical framework for research in the Foundations of
Mathematics (FoM), Harvey a few posts back suggested that it is clear
that nothing much weaker than (full) FOL will do. One obvious
weakening would be to FOL with only a limited number of distinct
variables allowed: 3-variable FOL being, for example, equivalent in
expressive power to the "Algebra of Relations." Interestingly, a
moderately strong THEORY can compensate for the weakening of the
LOGIC: Tarski and Givant's "Set Theory without Variables" shows that
ZFC (and any of a wide variety of other theories of FoM-interest in
which pairing functions are definable) can be formulated in
3-variable FOL.
--
Allen Hazen
Philosophy Department
University of Melbourne
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