[FOM] N vs. FOL

Dana Scott Dana_Scott at gs2.sp.cs.cmu.edu
Thu Jun 5 13:41:32 EDT 2003

Vladimir Sazonov wrote (in part):

  >> Say, PA consists of several axioms and one axiom schema. It is
  >> based on FOL based on several (schematic) rules. We easily
  >> understand how to use these rules.  All of this is quite
  >> concrete, unlike N, which is both illusive and not determined
  >> enough (as ANY illusion).

In order to understand fully schematic rules, one has to understand
syntax.  A theory of syntax (just as arithmetic) could be based on
finite strings of 0s and 1s under concatenation.  Call this domain B
(for "binary").  I say, B and N amount to the same thing.  

Now for PARTICULAR strings (formulae) I can see how to operate on them
following the rules of FOL -- just as I can see how to use the rules
of arithmetic to operate on certain particular numbers met in everyday
life in the way we learn in school.  I agree these particullar
operations are "concrete".  However, for any GENERAL RESULTS, say even
for the Deduction Theorem, I need some kind of induction principle to
argue that some desirable property holds for ALL (provable) formulae
-- just as I need induction to prove in PA that, say, addition is
associative and commutative.

Yes, I have gone up to the metalanguage here in discussing the
formalization of theories.  But if B is "illusive and not determined
enough", what justifies my using these syntactical arguments?

I sense an infinte regress.

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