Joao Leao wrote:
>Jesse Mazer wrote:
>
> > As I think Bruno Marchal mentioned in a recent post, mathematicians use
>the
> > word "model" differently than physicists or other scientists. But again,
>I'm
> > not sure if model theory even makes sense if you drop all "Platonic"
> > assumptions about math.
>
>You are quite right! The answer is: it doesn't. Model Theory, in which
>Tarsky
>built a workable notion of truth is as subject to Godel Incompleteness as
>any
>other system of of axioms beyond a certain size. Basically the only
>mathematical
>models that do not suffer from this problem are isomorphic to binary
>boolean
>algebra of classes (though Set Theory suffers from its own problems).
Actually, I probably shouldn't have used the term "model theory" since
that's a technical field that I don't know much about and that may not
correspond to the more general notion of using "models" in proofs that I was
talking about. My use of the term "model" just refers to the idea of taking
the undefined terms in a formal axiomatic system and assigning them meaning
in terms of some mental picture we have, then using that picture to prove
something about the system such as its consistency. For example, the
original proof that non-Euclidean geometry was consistent involved
interpreting "parallel lines" as great circles on a sphere, and showing that
all the axioms correctly described this situation. Likewise, Hofstadter's
simple example of an axiomatic system that could be interpreted in terms of
edges and vertices of a triangle proved that that axiomatic system was
consistent, assuming there is no hidden inconsistency in our notion of
triangles (an assumption a Platonist should be willing to make).
On the other hand, here's a webpage that gives a capsule definition of
"model theory":
http://linas.org/mirrors/www.ltn.lv/2001.03.27/~podnieks/gta.html
"All these results have been obtained by means of the so-called model
theory. This is a very specific approach to investigation of formal theories
as mathematical objects. Model theory is using the full power of set theory.
Its results and proofs can be formalized in ZFC. Model theory is
investigation of formal theories in the metatheory ZFC."
I would guess that this means that to prove arithmetic's consistency in
model theory, you identify terms in arithmetic with terms in ZFC set theory,
like identifying the finite ordinals with the integers in arithmetic, and
then you use this to prove arithmetic is consistent within ZFC. However,
Godel's theorem applies to ZFC itself, so the most we can really prove with
this method is something like "if ZFC is consistent, then so is arithmetic".
Is this correct, and if not, could you clarify?
There would be no conditions on the proof of arithmetic's consistency using
my more platonic notion of a "model"--since we are certain there are no
inconsistencies in our mental model of numbers, addition, etc., we can feel
confident that Peano arithmetic is consistent, period. This may not be
"model theory" but it does involve a "model" of the kind in Hofstadter's
example.
Jesse Mazer
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Received on Mon Jun 16 2003 - 18:16:32 PDT