Re: Do we really nedd to have models for a theory?

From: David C. Ullrich (ullrich_at_math.okstate.edu)
Date: 03/26/05 

Date: Sat, 26 Mar 2005 07:43:38 -0600

On 25 Mar 2005 12:36:06 -0800, "george" <greeneg@cs.unc.edu> 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.
>
>Interesting that you begin that with "not understanding how Godel's
>Completeness Theorem is proven". It is only BECAUSE it has been proven
>that we know it is safe to do this. If we had started out without
>models, we would never have LEARNED this completeness theorem.
>
>> > At least
>> >it would seem we could view theorems that way without the necessity
>of
>> >introducing the concepts of models, would it not?
>
>It would only seem that way AFTER you have proved the completeness
>theorem.
>
>
>David C. Ullrich replied:
>> Yes, we could do that.
>>
>> We could also ignore logic entirely and spend our time on basket
>> weaving...
>
>This is an idiotic non sequitur.

Uh, no. You need to tune up the [not-sure-what-the-right-word-is-here]
detector: He asked whether we "could" ignore model theory, without
specifiying what we were trying to do. What we can and cannot ignore
depends on what our objective is.

>The whole point is that
>IF we did this, IF we started ignoring models, that would NOT AT ALL
>be like ignoring logic entirely. We would STILL BE DOING logic.
>That is, precisely as NDN has said, the import of the completeness
>theorem: IF ALL you care about is theorems, then YOU DON'T need models.
>We could ignore models altogether AND STILL BE CONCENTRATING on logic,
>and on the MORE IMPORTANT part of logic, at that.
>
>> One reason we care about models of a theory is that often it's
>> the model itself that we're interested in!
>
>Of course. But that is an argument IN FAVOR of ignoring logic
>altogether.
>If it is the model that you care about, then STICK to the model and
>IGNORE
>axiomatizing it or thinking about superclasses of other similar models.
>For NDN's benefit, however, you should simply stress that the model
>CAME FIRST, historically. We had numbers and addition and
>multiplication
>BEFORE we had axioms for Peano Arithmetic. We had triangles and ratios
>and equalities between them, and the Pythagorean result about right
>triangles,
>BEFORE we had Euclid's axioms. Historically we started with one
>preferred
>intended model and invoked inference-from-axioms (i.e., the creation of
>a
>formal theory, via logic) AS A TECHNIQUE to organize our knowledge
>about
>the model, as well as to make predictions about parts of it that were
>"harder" to observe. Spending a lot of time doing "logic for its own
>sake" tends to cause one to forget that the original PURPOSE of the
>theory
>was to expand knowledge ABOUT the model.
>
>> 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.
>
>Definitely the better reason.
>Logic is fine for confirming that something IS a theorem, but
>how do you confirm that it isN'T? The big sexy results of the
>2nd half of the 20th century in meta-mathematics were independence
>results (C independent of ZF, CH indpendent of ZFC), so that certainly
>reinforced "presenting a model" as a technique. But there is a real
>problem with that: The Model Description Language.
>How do you ever say which model you mean? What are models MADE OF,
>anyhow? People usually use ZFC but even that was obviously
>insufficient
>for some of the results ABOUT ZFC. And who's to say that using ZFC
>isn't
>somehow cheating, somehow lensing what you see?

************************

David C. Ullrich

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