Re: True = [ proven | provable ]
From: Lysander (lysander_at_hellas.net)
Date: 01/17/05
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Date: Mon, 17 Jan 2005 09:46:20 -0800
D.McAnally@i'm_a_gnu.uq.net.au (David McAnally) writes:
> Ogie Ogelthorpe <boogieloogie@gmail.com> writes:
>
>>|-|erc wrote:
>>> Mathematicians don't need the word true.
>>>
>>> For "I think its true" say "I think its provable".
>>>
>>> For "G is true" say "G is proven"
>>>
>>< snipped the rest of the useless drivel>
>
>>The only thing true is that you are a certified nut job who should be
>>locked up before you hurt yourself or someone else.
>
> The distinction between "provable" and "true" is easy to demonstrate.
>
> A sentence is "provable" or "unprovable" for a given theory (a set of
> sentences). It is inappropriate to describe a sentence as being "true"
> or "false" for a theory.
>
> A sentence is "true" or "false" for a specific model (the truth value of
> a formula for a certain assignment of variables within a model is defined
> by recursion on the complexity of the formula, and the truth value of a
> sentence for a model is independent of the assignment of variables).
> It is inappropriate to describe a sentence as being "provable" or
> "unprovable" for a model.
>
> So a sentence is "provable" or "unprovable" for a theory, but not for a
> model. A sentence is "true" or "false" for a model, but not for a theory.
>
> A sentence which is provable in a theory is true in all models of the
> theory.
>
> A sentence which is unprovable in a theory is false in some model(s) of
> the theory (i.e. it is false in at least one model of the theory).
>
> A sentence which is true in all models of the theory is provable in the
> theory.
>
> For the sentence which is used in the proof of Godel's Incompleteness
> Theorem, the interpretation given in the proof is the interpretation in
> the STANDARD MODEL of the natural numbers. It is NOT the interpretation
> in all models (i.e. the given interpretation is MODEL dependent). The
> sentence is unprovable in the theory of formal arithmetic, and it is true
> in the standard model. This does not cause a contradiction, since there
> are models of formal arithmetic in which the sentence is false, and in
> NONE of these models is the interpretation that given in the proof of the
> Incompleteness Theorem.
I don't necessarily disagree with anything you say, I'm just trying
to figure out exactly what you mean.
First off, my understanding (which may be seriously flawed) of the
Incompleteness Theorem is that Goedel, using construction rules
which are legitimate in the Principia Mathematica (PM), constructed
a statement which was true (on meta-mathematical grounds) but
formally unprovable within the system. The construction method uses
factorization, so the logical requirement is for the Fundamental
Theorem of Arithmetic, and, therefore,the proof only applies to
logic systems sufficiently powerful to use that theorem. Is this
what you mean by the 'STANDARD MODEL of the natural numbers, that is
natural numbers which we can decompose by factoring? And
what other models of the natural numbers are there?
According to your 'a sentence which is unprovable in a theory is
false in ... at least one model of the theory.' Which 'model' for a
PM like logical system would you propose which makes the statement
used in the Incompleteness Theorem false?
Just asking, not arguing that such doesn't exist.
L.
-- Prediction is difficult, especially the future. Niels Bohr
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