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Stream: deprecated: logic

Topic: roots as models

Morgan Rogers (he/him) (Feb 21 2023 at 22:02):

In joint work with @Damiano Mazza, a connection between algebra and logic is emerging. Since I'm sure I've seen it somewhere before but can't place the location, can anyone give us a reference of a situation where someone has viewed roots of polynomials as models of a theory, either formally or informally? I thought this might be in the work of someone like Hrushovski, but his papers are hard to navigate in searching for this stuff, and I'm really looking for a more basic/intuitive connection that I might have seen before.

Jean-Baptiste Vienney (Feb 21 2023 at 22:06):

Don't know but looks very interesting. Models of which kind of theories are you thinking about? Do you mean a connection between algebra and first order classical logic?

Jean-Baptiste Vienney (Feb 21 2023 at 22:09):

I'm interested by connections between algebra and logic but in a very different way (algebra and linear logic). I don't know anything about model theory that's why I'm interested to know a little more about what you mean.

Jean-Baptiste Vienney (Feb 21 2023 at 22:10):

I've read phrases like "model theory is the second one algebraic geometry" but have no idea what it means.

Jean-Baptiste Vienney (Feb 21 2023 at 22:15):

(Sorry not to help you. I'm just interested to be educated.)

John Baez (Feb 21 2023 at 22:25):

Isn't a root of a polynomial just the same as a model of the theory of the complex numbers (or your favorite commutative ring) together with a constant symbol $x$ and some axiom like $x^3 - 4x^2 + x = 0$?

John Baez (Feb 21 2023 at 22:29):

More precisely: you could do this in classical first-order logic using the axioms for an algebraically closed field of characteristic zero, and then a model would be a choice of some such field together with a root of the polynomial. But since the theory of an algebraically closed field of characteristic zero has just one model (up to isomorphism) for each uncountable cardinality, this is just about as good as what I said.

John Baez (Feb 21 2023 at 22:29):

Or, you could implement this idea in various other forms of logic.

John Baez (Feb 22 2023 at 00:36):

For example, you can think of a scheme as a theory and its $k$-points for any commutative ring $k$ as models of this theory.

John Baez (Feb 22 2023 at 00:38):

This includes the case of roots of polynomials: to recover my example above, take the affine scheme corresponding to the commutative ring $\mathbb{Z}[x]/\langle x^3 - 4x^2 + x \rangle$ and take $k = \mathbb{C}$.

John Baez (Feb 22 2023 at 00:39):

This stuff about schemes then eventually leads to Grothendieck's ideas on topos theory....

Damiano Mazza (Feb 22 2023 at 08:40):

John Baez said:

For example, you can think of a scheme as a theory and its $k$-points for any commutative ring $k$ as models of this theory.

Yes, this is exactly the idea. This analogy with logic is clear and is probably "folklore". But I am not aware of anyone having worked it out to the extent of finding a common structure of which schemes (maybe only affine schemes) and logical theories (of some suitable fragment of first-order logic) are particular cases.

Damiano Mazza (Feb 22 2023 at 08:47):

Just to be clear: this is not about axiomatizing ring theory (or specializations of it, like algebraically closed fields of characteristic zero) in first-order logic. This is about finding some kind of "generalized logic" such that commutative ring presentations, given in terms of generators and polynomial equations, and classical first-order theories of some restricted kind (e.g. what Johnstone calls "Cartesian theories") presented in terms of sorts, relation symbols and axioms, are both instances of "generalized theories" for this "generalized logic".

Morgan Rogers (he/him) (Feb 22 2023 at 08:56):

(I should credit @Ivan Di Liberti for reminding me about Hrushovski via strongly minimal theories)

Damiano Mazza (Feb 22 2023 at 08:58):

For example, in this view, the set-theoretic completeness theorem for whatever fragment of first-order logic that works for the analogy would correspond to the (weak) Nullstellensatz for a certain "generalized field", namely the one in which we define our countable set-theoretic models (I am restricting to countable models because, presumably, if we consider arbitrary models there will be size issues with this hypothetical "generalized field". By the downward Löwenheim-Skolem theorem, countable models are enough for Gödel's completeness to hold, and this will keep the size under control, as long as our "generalized theories" have countable syntax, of course...).

Damiano Mazza (Feb 22 2023 at 08:59):

I am unaware of anyone having discussed, even informally, this analogy between the Nullstellensatz and the completeness theorem.

Ivan Di Liberti (Feb 22 2023 at 09:32):

The idea that dualities and in general syntax-semantics adjunctions can be understood as completeness theorems via reconstruction of the theory from its models can't be really associated to anyone in my opinion.

I do not have much time to write today, but maybe a version of what you want could be in GENERAL AFFINE ADJUNCTIONS, NULLSTELLENSA ̈TZE, AND DUALITIES by Caramello, Spada and Marra.

Dylan Braithwaite (Feb 22 2023 at 09:40):

Damiano Mazza said:

I am unaware of anyone having discussed, even informally, this analogy between the Nullstellensatz and the completeness theorem.

This blog post by @Eigil Rischel (and the tweet linked within) talk about this analogy

Morgan Rogers (he/him) (Feb 22 2023 at 09:44):

It's credited to a tweet of @sarahzrf who was quite active here years ago but hasn't posted in a long time.

Damiano Mazza (Feb 22 2023 at 09:59):

Dylan Braithwaite said:

This blog post by Eigil Rischel (and the tweet linked within) talk about this analogy

Yes, that's it! Thanks. This neatly formulates the analogy in terms of an adjunction (I think this is also related to @Ivan Di Liberti's post, but I haven't yet read the paper he pointed to). So what I wrote above could be phrased by saying that there is one adjunction which covers, as special cases, both affine varieties and first-order logic (suitably restricted).

sarahzrf (Feb 22 2023 at 18:06):

who pings me

sarahzrf (Feb 22 2023 at 18:07):

oh lol im not sure i can be credited with anything like an original observation here

Graham Manuell (Feb 22 2023 at 22:32):

I'm not sure I completely understand what you guys are looking for, but maybe algebraic logic and/or the slogan "model theory = algebraic geometry - fields" are relevant? There are also these slides by Ingo Blechschmidt.

Morgan Rogers (he/him) (Feb 23 2023 at 09:01):

Thanks for the references @Graham Manuell !

Patrick Nicodemus (Feb 24 2023 at 02:09):

See tent and ziegler's introductory textbook for a model theoretic proof of the nullstellensatz. it's my understanding that more generally theories that have prime models satisfy a kind of nullstellensatz bur i don't remember the details.

Evan Washington (Feb 26 2023 at 04:36):

Spencer Breiner's thesis scheme representation for first-order logic and this later paper from his advisor Steve Awodey seem very relevant here. They develop the analogy with algebraic geometry pretty well, such that their "logical schemes" can behave analogously to the usual schemes