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Topic: Matheology § 224
Replies: 811   Last Post: Apr 14, 2013 1:28 PM

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 fom Posts: 1,968 Registered: 12/4/12
Re: Matheology § 224
Posted: Apr 12, 2013 9:17 PM

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.

Date Subject Author
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