On Nov 27, 2007, at 11:03 AM, Pat Hayes wrote:
> [Rick Murphy wrote:]
>> ...I remain unconvinced that there's no connection between what
>> Feferman's writing about and CL.
>
> I don't know how to convince you without you first taking a graduate
> course in foundations of mathematics, but if you have the above
> reference right, perhaps you could just take it on authority? They
> really have nothing at all to do with one another. (01)
Right. (02)
> (The IKL project, now, may indeed have some possible relationship to
> Feferman's stuff, since IKL can define its own truthpredicate.) (03)
Yes, indeed. If axioms are added to IKL that bestow a "finegrained",
sentencelike structure on IKL's propositions, one could probably
reframe a lot of what Feferman is doing in terms of IKL. Feferman
works in an extension of Peano Arithmetic, after all, just so that he
can arithmetize the syntax of his theory. By quantifying over Gödel
numbers of sentences (and the other syntactic objects) of his theory,
the theory, in effect, is its own metalanguage, but without all the
messiness and complications required in a more standard sort of
metalanguage. (04)
Rather than quantifying over Gödel numbers of sentences, IKL
quantifies over propositions *expressed* by sentences. And since
there is, in effect, a truth predicate in IKL, many of the same issues
that Feferman is addressing can arise in IKL. (05)
BTW, Rick singles out McGee and Feferman, but they are but two stars
in a vast firmament here. Their (very fine) work is just a small
portion of the huge body of research on "postTarskian" semantic
theories for languages that contain their own truth predicate,
stemming (most notably) from Kripke's enormously influential 1975
paper "Outline of a Theory of Truth." (06)
>>>> Interesting, thanks for the info, anything you could refer me to so
>>>> I can read up on this ?
>>>
>>> On LBase as a precursor to Common Logic? On the incompleteness
>>> (better, perhaps, incompleteability) of arithmetic? On the
>>> completeness of firstorder logic?
>>
>> Completeness of FOL.
>
> Read any logic textbook that has a chapter on semantics or model
> theory. (07)
Many good choices here, but two I often recommend are Mendelson's
_Introduction to Mathematical Logic_ and Enderton's _A Mathematical
Introduction to Logic_. A somewhat less expensive and more
approachable text is Christopher Leary's _A Friendly Introduction to
Mathematical Logic_. The latter is out of print, but you can find
used copies on Amazon. (08)
chris (09)
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