FOM: Re: Re: finite axiomatization and conservative...

[Peter Apostoli]

Dear Allen,

Please help me out here:

> <<<This is sort of a converse to Craig's Theorem, that the restriction to
a
> sublanguage of any axiomatized system has a, perhaps non-finite,
> axiomatization IN the sublanguage.>>>

I am a little confused; is the "phil sci version of Craig's Theorem" you
cite just the Joint Consistency Theorem (?) (an equivalent of Craig's
interpolation Lemma). What you go on to call Craig's "offical" theorem
[i.e., Any r.e. theory is recursively axiomatizable in first-order logic
(shown by a construction that doesn't add to the language)] is usually
expressed as the folklore-ish result that any axiomatizable (i.e, r.e.) f.o.
theory is recursively axiomatizable, and, if I am not mistaken, was first
proved in Lob's (or was it Henken's?) thesis (??) Perhaps you are right that
is was proved by Craig. Sorry if I haven't got my folklore straight.

> Note that the Craig's "official" theorem-- what I gave above is the
"philosophy of
> science" version about doing without theoretical terms-- is: any r.e.
> theory is recursively axiomatizable in first-order logic (shown by a
> construction that doesn't add to the language).  Kleene's theorem
> strengthens this by changing "recursive" to "finite," but has to accept
new
> predicates.)
> Allen Hazen
> Philosophy Department
> University of Melbourne
> (interested in logic, logical metaphysics, philosophy of mathematics,
etc.)
>

Cheers

Peter A.
Philosophy
Univ. of Tor.
(inerested in foundations of mathematics)