Pat and Alex, (01)
PH
> All of modern mathematics, for example, can be written in FOL. (02)
AS
> Let me cite from Handbook of Mathematical Logic, J. Barwise...
> "2.6. There is no set of FOL axioms to describe R [real numbers]
> isomorphically." (03)
PH
> Isomorphically, of course. There is no set of FO axioms to describe
> the natural numbers isomorphically, never mind the reals...
> what it boils down to is that the semantic and syntactic ideas
> of truth ... must be representable as a FO theory. (04)
These statements require some qualifications. I scanned the beginning
of Section 5 "Beyond first-order logic", p. 41 of that same article.
Note the last sentence of the attached file, Barwise77.pdf:
> It would be completely impractical and, in fact counterproductive,
> to always make all one's extra-logical assumptions explicit. (05)
Barwise discusses six kinds of logics that are more "practical"
for some purposes: 5.1 many-sorted first-order logic; 5.2 ω-logic;
5.3 weak second-order logic; 5.4 infinitary logic; 5.5 logic with
new quantifiers; abstract model theory. (06)
Common Logic is an example of a very general extension of FOL that
allows quantified variables to refer to functions and relations.
But it's important to provide some guidance to show how to use CL
to support the many versions of logics used in applications. (07)
John (08)
Barwise77.pdf
Description: Adobe PDF document
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