Re: model theory: What's the big picture?

On Thu, 02 Apr 2009 20:07:40 -0700, Ben Crowell
<crowell09@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:

I'm reading A Shorter Model Theory, by Hodges, and have gotten through
the first couple of chapters. So far I'm having a very hard time
figuring out what's the main thrust of the subject. For comparison,
suppose I know someone who is going to teach himself calculus. He
wants to know what it's about. I'd say, "Calculus is about rates of
change. If you have a function f that changes, you can find another
function g that measures f's rate of change. Going backward, if you
know g you can find f, which tells you how much change accumulates.
This relationship between f and g is summarized by something called
the fundamental theorem of calculus. As you're learning calculus,
you'll spend most of your time learning technical stuff about how
to go from f to g and from g to f."

Could anyone give me a similar broad-strokes overview of what model
theory is about?

What's missing for me so far is the feeling of "Aha! Here's something
I wanted to do, and now I know how to do it using model theory."

Not to say that it's _the_ point, but one big point to model theory
is showing that some set of axioms is _consistent_.

The closest I came to that so far is his example of adjoining the root
of an irreducible polynomial to a field, e.g., i. First you form a field
of polynomials F[i] in the indeterminate i, and make T be the set of all
true atomic formulas for F[i]. Then you just add one more formula to T,
stating that i^2=-1, to make T'. The canonical model of T' has i^2 and
-1 in the same equivalence class. Okay, that's kind of cool.

Here, for example, we have field theory, we have the theory of T,
and we want to be certain that if we assume in addition that i^2 = -1
that will not lead to any inconsistency. Someone else might want
to add the axiom 0 = 1, but that person would be out of luck.
We show that our original theory plus i^2 = -1 is coniststent
by showing it _has_ a model.

A more realistic example: We're doing mathematics some time
ago, and when we get around to being very careful and formal
about our assumptions we notice that we've been using the
Axiom of Choice all the time. No problem, since AC is
obviously true, just pick one element from each set.

Then we notice that AC gives us the Banach-Tarski "paradox",
and while we realize that there's nothing literally contradictory
about B-T it certainly seems very strange. So we're worried
that assuming AC is going to lead to a contradiction, making
our universe explode. Then along comes Godel, who proves

(*) if ZF (set theory )is consistent then ZFC (ZF + AC)
is consistent.

And so we relax about using AC - we don't actually _know_
that ZFC is consistent, but we do know that if we do
someday get a contradiction it's not AC's fault, there
was already a contradiction in the rest of set theory.

So (*) is a good thing. And how do we prove (*)?
By assuming that ZF has a model, and using it to
construct a model of ZFC.


On the
other hand, the construction is trivial if you simply define complex
numbers as ordered pairs, so this fails to fall into the category of
something I wouldn't have known how to do without knowing model theory.

David C. Ullrich

"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
.

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