Re: True = [ proven | provable ]
From: David McAnally (D.McAnally_at_i'm_a_gnu.uq.net.au)
Date: 01/17/05
- Next message: Mike Oliver: "Re: computation and cardinality"
- Previous message: Mike Oliver: "Re: computation and cardinality"
- In reply to: Ogie Ogelthorpe: "Re: True = [ proven | provable ]"
- Next in thread: Lysander: "Re: True = [ proven | provable ]"
- Reply: Lysander: "Re: True = [ proven | provable ]"
- Reply: |-|erc: "Re: True = [ proven | provable ]"
- Messages sorted by: [ date ] [ thread ]
Date: 17 Jan 2005 14:48:02 GMT
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.
David
-----
- Next message: Mike Oliver: "Re: computation and cardinality"
- Previous message: Mike Oliver: "Re: computation and cardinality"
- In reply to: Ogie Ogelthorpe: "Re: True = [ proven | provable ]"
- Next in thread: Lysander: "Re: True = [ proven | provable ]"
- Reply: Lysander: "Re: True = [ proven | provable ]"
- Reply: |-|erc: "Re: True = [ proven | provable ]"
- Messages sorted by: [ date ] [ thread ]
Relevant Pages
|
