Re: Do we really nedd to have models for a theory?
From: David C. Ullrich (ullrich_at_math.okstate.edu)
Date: 03/26/05
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Date: Sat, 26 Mar 2005 07:41:15 -0600
On Fri, 25 Mar 2005 18:09:45 GMT, namducnguyen <namducnguyen@shaw.ca>
wrote:
>
>
>David C. Ullrich wrote:
>
>>On Thu, 24 Mar 2005 05:55:47 GMT, namducnguyen <namducnguyen@shaw.ca>
>>wrote:
>>
>>
>>
>>>Not understanding how Godel's Completeness Theorem is proven,
>>>I wonder if models are necessary for a 1st order theory? I mean to the
>>>extend that theorems are true in all models, why would we care about
>>>models at all? If we want theorems, we could simply prove them - produce
>>>them -
>>>mechanically by inferring them from axioms using rules of inference. At
>>>least
>>>it would seem we could view theorems that way without the necessity of
>>>introducing the concepts of models, would it not?
>>>
>>>
>>
>>Yes, we could do that.
>>
>>We could also ignore logic entirely and spend our time on basket
>>weaving...
>>
>>One reason we care about models of a theory is that often it's
>>the model itself that we're interested in! Another reason is
>>that the standard way of showing that P does _not_ follow from
>>a set of axioms A is to construct a model of A in which P is false.
>>
>>
>
>Then again, there are counter examples to the utility of models that would
>seem to prompt questions about their necessity.
>
>Example1: Let T be a theory in which Phi(x) is known to be undecidable by
> a non model-theoretic method.
Can you give an example of such a T and Phi?
(An example that people actually care about, not just one contrived
to answer the question.)
> Let A be an axiom of T. We'd
> know immediately that Phi'(x) = (A /\ Phi(x)) is undecidable,
> without the necessity of knowing which model of T Phi'(x)
> is true, or false.
>
>Example2: It would take some work but the Chess game (the Chinese version,
> or the Western version) can be thought of as a 1st-order
> formal theory. But in playing the game, are we really interested
> in a model the Chess theory? It seems that each move from us
> or from the opponent would be just another hypothesis added in
> to a long chain of hypotheses, which eventually lead to a
> certain conclusion. I wouldn't think chess players would care
> much about models of the Chess game!
You didn't bother to read my entire post.
This is an example of the second situation I mentioned: Chess players
don't care one bit about that formal theory, what is of interest
to chess players is one specific model, namely a chess board.
************************
David C. Ullrich
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