As of May 31, 2023, we have updated our Code of Conduct.

# Questions tagged [model-theory]

Model theory is the branch of mathematical logic which deals with the connection between a formal language and its interpretations, or models.

1,101 questions
Filter by
Sorted by
Tagged with
1 vote
73 views

### Extending a first-order deductive system with satisfaction relation

I'm trying to structure a proof where there are several algebras instantiated over sets, where the properties that you get from the algebraic theories are important, but the properties of the sets ...
1 vote
71 views

### Lifting Wilkie's theorem from $\mathbb{N}$ to other structures

Let $\models,\models_2,\models_d$ be the satisfaction relations for first-order logic, second-order logic with full semantics, and second-order logic with set quantifiers ranging over definable-with-...
1 vote
117 views

### 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 ...
328 views

### The scope of a "strong Cantor-Bernstein" property

This question is of course related to this earlier MO question, but I don't believe is answered by the posts there. My favorite proof of the Cantor-Schroeder-Bernstein theorem actually establishes ...
133 views

79 views

### Counterexamples to the definable (P,Q)-Theorem

Pierre Simon conjectured a model-theoretic definable version of Matoušek’s (p,q) -theorem in NIP theories: [Conjecture 5.1]: Let T be NIP and M⊨T . Let ϕ(x;d) ∈ L(U) a formula, non-forking over M . ...
104 views

### What logics do the transfinite length pebble games capture?

See e.g. Libkin, Elements of finite model theory for background on the usual pebble game. Below, all languages are finite and relational, and "$\uparrow$" denotes an expression being ...
311 views

### Various sizes of models of NBG inside NBG (what does a class-sized model give us?)

Following this question and that one illustrating how induction in NBG can be tricky, I realized I'm also confused about the notion of “model” of NBG. The goal of this confusion is to hopefully lift ...
144 views

### Some questions about the Hyperuniverse Program

The Hyperuniverse Program, founded by Sy D. Friedman, intends to produce new second-order axioms of set theory which appropriately formalize "the universe is maximal" in one of a few ways. A ...
206 views

### Does every ordinal appearing in a model of ZF appear in a model of ZFC?

To be more precise, suppose that $M$ is a model of ZF, for simplicity (or tactility) a set in some larger model $V$ of ZFC+Con(ZF), and suppose that $M\vDash\alpha$ is an ordinal$"$. Must there be a ...
159 views

### When does HSP reduce to SPH?

This is actually a poorly camouflaged attempt to use the answers to When is the opposite of the category of algebras of a Lawvere theory extensive? (all very interesting) for the purposes of my ...
152 views

### Is there a largest o-minimal structure all of whose definable functions are analytic?

In the paper "Quasianalytic Denjoy-Carleman classes and o-minimality" by J.-P. Rolin, P. Speissegger and A. J. Wilkie, it is proven that there is no largest o-minimal structure on the real ...
148 views

### Need proof on a model being elementarily equivalent but non-isomorphic

For countable models, elementary equivalence is not equivalent to isomorphism. For example, let $\frak{A}= \omega+\Bbb{Z}*\omega$ and $\frak{B}= \omega+\Bbb{Z}*\omega^*$ ($ω^∗$ is the reverse of $ω$)... 113 views

### Is there an ordered algebra analogue of the HSP theorem?

For an algebraic signature (= set of function symbols) $\Sigma$, say that an ordered $\Sigma$-algebra is a pair $\mathfrak{A}=(\mathcal{A};\le)$ where $\mathcal{A}$ is a $\Sigma$-algebra in the sense ...
64 views

143 views

### Is having a Frobenius pair first-order expressible in the language of groups?

I am trying to figure out whether or not the following property is first-order expressible in the language of groups. $$\text{G has a subgroup H with which it forms a Frobenius pair (H,G).}$$ My ...
111 views

### Does the "iterated definability" closure always fall short of standard (Boolean) infinitarization?

Given a (set-sized) logic $\mathcal{L}$, let $\mathcal{L}'$ be gotten from $\mathcal{L}$ by adding the ability to quantify over $\mathcal{L}$-definable relations. (I'm being a bit vague here, since e....
256 views

### Modal logic of "mostly-satisfiability"

For $n\in\omega+1$ let $\mathsf{ZFC}_n$ be $\mathsf{ZC}$ + $\{\Sigma_k$-$\mathsf{Rep}: k<n\}$. Let $\widehat{\mathsf{ZFC}}$ be the strongest consistent theory $\mathsf{ZFC}_n$ (so if $\mathsf{ZFC}$ ...
119 views

### Systems of elementary embeddings

Recently I've been thinking about elementary embeddings, partition cardinals, etc. as part of my never-ending quest for understanding of consistency strength :p I came up with this idea, called I* ...
67 views

193 views

### Does $\mathit{Aut}(\mathbb{R};+)$ have a copy in $L(\mathbb{R})$ granting large cardinals?

Throughout, work in $\mathsf{ZFC}$ + large cardinals (let's say a proper class of Woodin limits of Woodins but I'm happy to go higher if that would help). Let $\mathcal{R}=(\mathbb{R};+)$ be the ...
87 views

138 views

### Does every Tarski plane embed into a 3-dimensional Tarski space?

By a Tarski space I understand a mathematical structure $(X,B,\equiv)$ consisting of set $X$, a betweenness relation $B\subseteq X^3$ and a congruence relation ${\equiv}\subseteq X^2\times X^2$ ...
455 views

240 views

### Is this notion of "concrete bijection" transitive?

This question looks at the same intuition as, but expressed via a different formal notion than, a couple earlier questions of mine (1, 2). Basically, I'm playing around with using model theory to ...