On 4/12/2013 6:31 PM, Frederick Williams wrote: > Nam Nguyen wrote: > >> In any rate, "proved true in all [formal] systems" is a mixed-up >> of technical terminologies: formal systems prove syntactical theorems, >> truths are verified in language structures. The two paradigms are >> different and _independent_ : proving in one doesn't logical equate >> to the other. > > G\"odel's completeness theorem shows that they aren't independent. >
That is actually a good thing, although I never recognized it until I saw something Kleene wrote.
An instantiated counterexample must be able to defeat a bad proof. One cannot prove a counterexample. The model theory introduces this aspect.
I now see one thing in Nam's statement I did not notice before.
It is not that truths are verified. It is that meaning is defined.
The typical sense of the Fregean analysis is that the meaning of a statement is derived from its ground with reality. In this, the referent of the statement is to act as that ground. Frege introduced "The True" and "The False" specifically as object references for propositions just as material objects may be taken as reference for terms like "the dog" or "the cat".
But, Frege introduced these notions prior to the formalist program. The model-theoretic sense of truth in relation to formalism need not be the realist philosophy Frege had when developing his logic. That, too, is lost when textbook authors do not explain their exposition with a historical context.
One may object to the paradigm and offer a new one. But that is its definition.
It is likely that Nam's notion of verification corresponds with the mathematics of the Russian school of constructive mathematics.