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Does anything interesting happen to model theory when we move from logic+set theory to type theory (including the truncated part which is logic)?
I mean what happens to a book like Chang and Keisler’s classic Model Theory? Can we go through the introduction changing things systematically?
So p. 1:
universal algebra + logic = model theory
To arrive at model theory, we set up our formal language, the first-order logic with equality.
p. 2
Löwenheim’s theorem, the compactness theorem, and Morley’s theorem are
…typical results of model theory. They something negative about the ’power of expression of first-order predicate logic…
but
…also say something positive about the existence of models having certain properties.
Etc.
I think, to be relevant for the present state of the art in model theory, we should go beyond first order. For example, to include abstract elementary classes.
The article abstract elementary class needed some cleaning up, and I tried to do that, but you (Zoran) should probably take a look when you have a moment, to make sure I didn’t introduce errors.
I wish I knew how to help, but it looks like all model theory entries need an upgrade. I find the array of category-theoretic accounts of model theory very confusing.
@David: there are five nLab articles with “model theory” in the title (model theory, stability in model theory, structure (model theory), structure in model theory, type (in model theory)), and many more where “model theory” is a string in the page text. Can you say which entries were confusing, or do any particularly stand out?
It’s not so much confusion, as incompleteness, e.g., stability in model theory. But maybe what I’m after is a more category theoretic approach to the subject. The dissatisfaction arose from my brief foray into how Tao should be answered. There seem to be a range of ways to do so, and I have no means to rate them or to compare them.
What’s the difference between structure in model theory and structure (model theory)?
I probably share your wishes, David (and was disappointed when I tried to click on Makkai’s survey referenced in stability in model theory, only to find that the doi link is broken). It seems to me that many of the really knowledgeable people on the interface between model theory and category theory – thinking particularly of Makkai here although others also merit mention: Paré, Gonzalo Reyes, Moerdijk, et al. – are not active in the blogosphere. It seems one might have to talk with them directly.
It’s really a regrettable situation that so few category theorists seem to have deep knowledge of model theory, because it’s obviously immensely powerful in the hands of those who know how to wield it. One of my “bucket projects” is to acquire some non-trivial understanding of stability theory. What meager attempts I’ve made tend to make me sympathetic towards Kazhdan’s descriptions.
Still, I’d like to encourage you to keep talking. Would you be willing to expand on this?
There seem to be a range of ways to do so, and I have no means to rate them or to compare them.
Google helped me fix the DOI link at stability in model theory.
Thanks, Toby! Too bad it’s a blasted Springerlink.
Todd, I’m sure you know far more about model theory than I do. There appear to be quite a number of people working at the interface, aside from the category theorists you list, e.g., Mike Prest and Moshe Kamensky, both appearing in a book in honour of Makkai, and Michael Lieberman, see, e.g., these slides.
Then there’s the approach via institutions.
As a side note:
I have just added a bunch of links to keywords at model theory. More links or different links might be desireable. I’d like to ask experts to consider looking into this. Because if we assume that a reader of this entry does not yet know model theory, we should assume that he also does not know all the terms “language, signature, structure, truth, etc.” in this context, and so explaining one by the other without further hyperlinks is likely not to come across as an explanation.
In this context I can offer a first-hand impression of that entry, in case anyone into model theory finds this a useful feedback:
currently the entry leaves the reader pretty much entriely unclear about why anyone would bother about model theory, especially if that somebody already knows what algebras over operads and Lawvere theories are.
What is missing from the entry, I think, are some paragraphs that say: model theory is great because … it has led to considerable progress when applied to … from model theory are motivated the important concepts …
Something like this.
That’s a great idea, Urs.
I have tried to write an Idea section for stability in model theory. It is not based on personal acquaintance; it was basically taken from Cherlin’s review of Baldwin’s book. Hopefully someone more knowledgeable will step forward and add more.
What’s the relation to geometric stability theory? I guess I once sat in a talk by Boris Zilber, but I seem to forget.
I’m not really able to say yet. But I’m looking now at some nice notes by Zilber.
I’ve begun to add a little to geometric stability theory.
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