All Questions
Tagged with model-theory reference-request
120
questions
17
votes
6
answers
2k
views
Book recommendation introduction to model theory
Next semester I will be teaching model theory to master students. The course is designed to be "soft", with no ambition of getting to the very hardcore stuff. Currently, this is the syllabus....
- 8,475
1
vote
0
answers
68
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Finitely presentable groups are residually finite if and only if they are universally pseudofinite
Suppose $G$ is finitely presentable. Then residual finiteness of $G$ is equivalent to $G$ satisfying the universal theory of finite groups (equivalently, to every existential statement true in $G$ ...
- 1,248
5
votes
0
answers
227
views
Classical first-order model theory via hyperdoctrines
I have been reading this discussion by John Baez and Michael Weiss and I find this approach to model theory using boolean hyper-doctrines very interesting. One of their goal was to arrive at a proof ...
- 3,084
6
votes
0
answers
165
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Elementary equivalence for rings
Let $\mathcal{L}$ be a first-order language, and $M$ and $N$ be two $\mathcal{L}$-structures. We say that $M$ and $N$ are elementarily equivalent (write $M \approx N$) if they satisfy the same first-...
- 3,104
1
vote
1
answer
239
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"On models of elementary elliptic geometry"
While perusing p. 237 of the 3rd ed. of Marvin Greenberg's book on Euclidean and non-Euclidean geometries, I learned that it can actually be proven that "all possible models of hyperbolic ...
- 8,762
6
votes
0
answers
215
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Reference for Understanding Shelah's Proof of Vaught's Conjecture for $\omega$-stable Theories
I'm looking for a source to help me better understand Shelah's proof of Vaught's Conjecture for $\omega$-stable Theories (https://shelah.logic.at/files/95409/158.pdf). An obvious candidate is Makkai's ...
- 173
2
votes
0
answers
162
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Which first-order theories have full indiscernible extraction?
Stable theories have the following useful property, which I will state in a sub-optimal way for simplicity's sake:
Fact 1. If $T$ is $\lambda$-stable for some $\lambda \geq |T|^+$, then for any set ...
- 10.4k
1
vote
1
answer
146
views
Stability theory in the context of $\omega$-stable theories
I'm looking for some references to get me started on stability theory. More specifically, I want to find sources that talk about notions in stability theory, but for $\omega$-stable theories, which ...
- 173
4
votes
1
answer
158
views
Source on equality-free second-order logic (nontrivially construed)
Throughout I'm only interested in the standard semantics for second-order logic, and all structures/languages are relational for simplicity.
If defined naively, second-order logic without equality is ...
- 20.7k
9
votes
0
answers
269
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Is “simplicity is elementary” still hard? (Felgner’s 1990 theorem on simple groups, and subsequent work)
I came across a reference in this MathOverflow answer to an intriguing result of Ulrich Felgner [1]: among finite non-Abelian groups, the property of being simple is first-order definable. According ...
- 19.6k
5
votes
1
answer
247
views
Free algebras from model theory perspective
Let $\mathbb{V}$ be a non-trivial variety of algebras, and let $F_S\in\mathbb{V}$ be a free algebra on a set $S$. I want to know what is known about the model theory of $F_S$; I know these objects are ...
- 51
4
votes
2
answers
646
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Tarski's original proof of quantifier elimination in algebraically closed fields
I am currently helping teach a course about foundations of mathematics, which has thus far focused mostly on propositional and first-order logic. As part of the course, the students are each required ...
- 741
6
votes
0
answers
237
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Existing literature on logics "describing their own equivalence notions"
Say that a regular logic $\mathcal{L}$ is self-equivalence-describing (SED) iff for every finite language $\Sigma$ there is a larger language $\Sigma'$ containing at least $\Sigma$ and two new unary ...
- 20.7k
4
votes
1
answer
420
views
Alternative proof of Tennenbaum's theorem
The standard proof of Tennenbaums's theorem uses the existence of recursively enumerable inseparable sets and is presented e.g. in Kaye [1, 2], Smith [3].
In the following, $\mathcal{M}$ will always ...
- 211
6
votes
1
answer
224
views
Sharp Craig interpolation theorem for $L_{\omega_1 \omega}$
I’d like to know if a sharp version of Craig’s interpolation theorem for $L_{\omega_1 \omega}$ is already known or exists in the literature. By a “sharp” version of this theorem, I mean something like ...
- 203