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
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":
"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.
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