# 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.

980 questions
Filter by
Sorted by
Tagged with
242 views

### How hard must "no high-degree irreducibles" proofs be?

Let $\mathsf{RCF}$ be the usual theory of real closed fields and for $n>2$ let $\theta_n$ be the sentence "No degree-$n$ polynomial is irreducible." Since $\mathsf{RCF}$ is complete, for ...
53 views

### Persistent finite axiomatizability, relational edition

Say that a finitely axiomatizable first-order theory $T$ (in a finite language $\Sigma$) is persistently finitely axiomatizable iff the set $T_{\mathsf{w/o=}}$ of equality-free $T$-theorems is ...
185 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 ...
1 vote
54 views

192 views

### Are equinumerous size preserving models of a theory isomorphic?

If by a size preserving model we mean any bijection between any two elements of it is an element of it. Then: is it a thoerem of $\sf ZFC$ that for any theory $T$ any two equinumerous size preserving ...
133 views

115 views

### When does $\Pi_2$-reflection on $X$ fail to imply iterated $\Pi_1$-reflection on $X$?

Let lowercase Greek letters denote ordinals. Recall from Richter and Aczel's "Inductive definitions and reflecting properties of admissible ordinals", for a set of formulae $\Gamma$ and a ...
191 views

### Transfinitely iterating the Levi-Civita, Hahn or Puiseux constructions

This question was originally asked at MSE but seems too advanced, so I'm reposting it here. In short, the idea is that many constructions for non-Archimedean fields can naturally be iterated, in some ...
318 views

### Is there a finitely axiomatizable class of structures whose equality-free theory is not finitely axiomatizable?

This was originally an MSE question, but I was told to ask it on MathOverflow. Does there exist a class $C$ of $L$-structures for a finite signature $L$, which is finitely axiomatizable in first-order ...
111 views

### Failure of Łoś-Tarski preservation theorem for some equality-free logic

A famous Łoś-Tarski preservation theorem is that first-order (FO) sentences that are preserved under substructures (resp. superstructures) are precisely the universal (resp. existential) first-order ...
222 views

169 views

### Why does the category of definable sets of $T^\mathrm{eq}$ have coproducts?

For each first-order theory $T$ there is an associated weak syntactic category, sometimes also called "the category of definable sets of $T$" and denoted $\mathrm{Def}(T)$. Also, for each ...
389 views

### A "surnatural numbers" as a largest model of the natural numbers

One characteristic of the surreal numbers is that they are a monster model of the first-order theory of real numbers, according to Joel David Hamkins in this post. Thus they are real-closed, and every ...
124 views

### The locale of morphisms vs a morphism to an ultrapower?

I'm fixing some type of structure $\Sigma$ (possibly multi-sorted, with functions symbols and relation symbols, though assuming it single sorted with only relation symbols wouldn't change anything). ...
340 views

### Do second-order theories always have irredundant axiomatizations?

It's a standard exercise to show that every countable first-order theory has an irredundant axiomatization. For uncountable first-order theories, the result is much more difficult and was proved by ...
258 views

### Can a non-standard model $M$ of $\mathsf{ZF}$ contain an internally infinite set that is externally smaller than $\aleph_0^M$?

Cardinalities in non-standard models of $\mathsf{ZF}$ are not generally reflected externally, but certain internal facts about cardinalities are always externally reflected (e.g., any internally ...
327 views

### Why is it so hard to give examples of differentially closed fields?

The theory of algebraically closed field, say in characteristic zero, and of differentially closed fields (of characteristic zero) have much in common: quantifier elimination and (hence) decidability; ...
219 views

### Is there a nice(r) counterexample to this strengthening of Tarski's theorem?

Given a regular logic $\mathcal{L}$ and a structure $\mathfrak{A}$ (in a finite relational language $\Sigma$ for simplicity), let $\overline{Th_\mathcal{L}(\mathfrak{A})}$ be the structure defined as ...
139 views

256 views

### Quantifier elimination in $S^1$

Does quantifier elimination (by cylindrical decomposition) work for systems of polynomial equations and inequalities where some or all of the variables are complex numbers of unit modulus, rather than ...
347 views

### Can $\mathcal{L}_{\omega_1,\omega}$ detect $\mathcal{L}_{\omega_1,\omega}$-equivalence?

Roughly speaking, say that a logic $\mathcal{L}$ is self-equivalence-defining (SED) iff for each finite signature $\Sigma$ there is a larger signature $\Sigma'\supseteq\Sigma\sqcup\{A,B\}$ with $A,B$ ...
308 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 . In the following, $\mathcal{M}$ will always ...
113 views

### Question on the model completeness of the real field expanded by restricted Pfaffian functions

Currently I'm reading "Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function" by Wilkie, and I do not ...
298 views

### Dehornoy's proof that the application of two elementary embeddings is an elementary embedding

What is meant by the statement and the proof of Lemma 3.2 in Chapter XII of Dehornoy's book Braids and Self-Distributivity? That lemma states "Assume that $j_1$ and $j_2$ are elementary ...
173 views

### A restricted form of the inner model hypothesis

Previously asked and bountied at MSE, with slight difference. To keep things relatively simple I'm presenting a somewhat-butchered version of the IMH; for more details, see S.-D. Friedman, Internal ...
197 views

### Can the forcing-absolute fragment of SOL have a strong Lowenheim-Skolem property?

Previously asked and bountied at MSE: Let $\mathsf{SOL_{abs}}$ be the "forcing-absolute" fragment of second-order logic - that is, the set of second-order formulas $\varphi$ such that for ...
199 views

### Does there always exist a categorical extension of $ZFC_2$ with no set models?

$ZFC_2$, i.e. second-order Zermelo-Fraenkel set theory with Choice, has only one proper class model upto isomorphism, namely $V$. But it may or may not also have set models. If $V$ has no ...
355 views

### Can there be no "surprisingly averageable" second-order sentences?

Say that a second-order sentence $\varphi$ is averageable iff there exists some infinite cardinal $\kappa$ and some nonprincipal ultrafilter $\mathcal{U}$ on $\kappa$ such that for every $\kappa$-...
Say that two structures $\mathfrak{A},\mathfrak{B}$ in possibly different finite languages are parametrically equivalent - and write "$\mathfrak{A}\approx\mathfrak{B}$" - iff they have the ...
Say that a logic $\mathcal{L}$ satisfies the weak test property iff for all $\mathfrak{A}\subseteq\mathfrak{B}$ we have $(1)\implies(2)$ below: For each $\mathcal{L}$-formula $\varphi$ with ...