Re: Goldbach Conjecture & the Foundation of First Order Logic.



Peter_Smith wrote:
> Nam Nguyen wrote:
>
>
>>But the problem in this case is that
>>for any given arithmetic model (of any domain) we could infer
>>the existence of *uncountably* other models (all are in the very same
>>domain). The imbalance between the "lesser" countable numbers
>>of languages and the "greater" uncountably numbers of models,
>>in the same domain, means that when we talk about a theorem of
>>"PA", we actually talk about infinite number of isomorphic
>>formulae, each of which is a theorem in the perspective isomorphic
>>theory.
>
>
> This is difficult to construe. But two points
>
> (1) Sure, there are an uncountable number of non-standard models for
> PA. But when we talk about *theorems* of PA we aren't talking
> semantics, we are talking about what follows from the non-logical
> axioms by the first-order deductive system built into PA.

Let me rephrase it differently. If we talk about *theorems* of a T, we
*don't need* to talk about semantics, but the semantics is already
there: provided of course T is *consistent*. The *only time* we aren't
- and couldn't be - talking about semantics is when T is *inconsistent*.
One would of course remember that every formulae "follows from the
non-logical axioms by the first-order deductive system built into" an
inconsistent T. But then good FOL reasoning is not supposed to be
based on an inconsistent formal system. So if we care about truth,
then semantics; model; and interpretation could not be disregarded.
And the complexity between models and isomorphic languages/theories
is still there.

> (There are no "isomorphic" theories in question -- there is the one
> theory, and we are interested in what you can and can't formally
> deduce in it.)

I think we all know what it means by Shoenfield's "A first-order
language is thus completely determined by its nonlogical symbols".
Therefore the 2 isomorphic languages

L'(0, + , <=) and
L''(0, U,included_in)

are different. Therefore the 2 isomorphic theories:

T' = {(Axy(x + y = y + x) /\ Ax(0 <= x))} and
T'' = {(Axy(x U y = y U x) /\ Ax(0 included_in x))}

are also different: and we know so because the lone axiom in T'
can be deduced from T' but not from T'', and vice versa.

> (2) When we talk about the *truths* of PA, again the uncountable class
> of non-standard models is irrelevant. As Torkel was fond of pointing
> out, you define truth in PA without mentioning models by a straight
> recursive definition.

and you cited a definition of truths in PA:

> Use | | for quoting formal language expressions. Define values for
> terms ...
>
> Val(|0|) = 0
> Val(|Sa|) = Val(|a|) + 1
> Val(|(a + b)|) = Val(|a|) + Val(|b|)
> Val(|(a x b)|) = Val(|a|) x Val(|b|)
>
> Atomic wffs are all equations between terms a, b. So we put
>
> |a = b| is True iff Val(|a|) = Val(|b|)
>
> Define prop. connectives as usual. Let Fx be context containing
> x free. Let *n* be SSSS..S0 with n occurrences of S.
>
> |ExFx| is True iff for some n, |F*n*| is true
>
> That's it! It assigns the correct truth-value to every wff of PA's
> language. End of story :-))

I wonder then what would be the truth values of ~GC, and GC? But
that's not the point. The point is since there are uncountably many
successor functions - each would correspond to one model out of
uncountable number of models (in the very same domain!) - we can't
assert in meta level that Val(|Sa|) = Val(|a|) + 1. In fact, we couldn't
even know precisely what Val(|0|) = 0 means, because the '0' on the
right side of "=" is a function of individual reasoner's interpretation
of what the S function be in the case; and, again, there are uncountably
many of them. (Remember, by default, we're assuming FOL=).

So, imho, it's far from being the end of the story; it actually seems
to be just a beginning, to see some problems in FOL reasoning!

--
-----------------------------------------------------

What we call 'I' is just a swinging door which moves
when we inhale and exhale.
Shunryu Suzuki
----------------------------------------------------

.

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