Model theory is the study of (classes of) mathematical structures (e.g. groups, fields, graphs, universes of set theory) using tools from mathematical logic. Objects of study in model theory are models for formal languages which are structures that give meaning to the sentences of these formal ...

learn more… | top users | synonyms

8
votes
3answers
514 views

Relationship between Category theory and Axiomatic set theory

I've recently started learning Category theory- and I have a pondering- wondering if anyone can help. Is it possible for two categories to satisfy two different set-axiom system. Namely- is it ...
5
votes
2answers
801 views

A first order sentence such that the finite Spectrum of that sentence is the prime numbers

The finite spectrum of a theory $T$ is the set of natural numbers such that there exists a model of that size. That is $Fs(T):= \{n \in \mathbb{N} | \exists \mathcal{M}\models T : |\mathcal{M}| =n\}$ ...
2
votes
1answer
415 views

some problems about axiomatizable classes resulting from compactness

I'm trying to work on the following problems: Prove that if a set S of sentences axiomatizes a finitely axiomatizable class K of structures then K can be axiomatized by a finite subset of S. I have ...
3
votes
1answer
124 views

does the definition of model depend on a theory or just a signature?

Σ:signature T:Σ-theory For M:Σ-model which satisfies T, is it proper to name it “Σ,T-model” or “Σ-model satisfying T”? In comparison, in the context of Lawvere theories, any Lawvere theory L ...
2
votes
0answers
70 views

Comparing symbolic and analog descriptions

I've never seen the following comparison before. Let me start with a specific example: Given a finite structure with two symmetric binary relations, i.e. a graph $G$ with one vertex set $V$ and two ...
3
votes
1answer
270 views

Class models in set theory and category theory

Is it a mistake ab initio to think of categories as of models of category theory, just as we think of (inner) models-of-set-theory as of models of set theory, graphs as of models of graph theory, ...
5
votes
2answers
534 views

Example of an UnDecidable Logical Theory which is an extension of a Logical Decidable Theory?

Let $T_1$ and $T_2$ be two first-order logical theories (over the same signature) such that $T_1 \subseteq T_2$ and both are recursively axiomatized. My question is the following: is it possible that ...
4
votes
1answer
333 views

(Un)Decidability of satisfiable but not finitely-satisfiable formulas

I am curios about the decidability or undecidability of the following decision problem. INPUT: a first-order formula $\varphi$ OUTPUT: Yes, if $\varphi$ is satisfiable but not finitely-satisfiable ...
1
vote
1answer
293 views

Finite Model Property on the First-Order Theory of Two Equivalence Relations

I know that there is a first-order sentence $\varphi$ such that $\varphi$ is written in the vocabulary given by just two binary relation symbols $E_1$, $E_2$ (and hence, without the equality ...
7
votes
2answers
246 views

Compactness on models with bounded finite size

I am aware that compactness fails on finite models, but the common counter-example uses models of arbitrary big finite size. So if we bound the size what results can we get? Assume we have an ...
5
votes
2answers
202 views

Logic in the metatheory

In Goldstern and Judah's The Incompleteness Phenomenon we are asked to prove that any model of the first two Peano Axioms: $$\forall x [Sx\neq0]$$ $$\forall x\forall y[Sx=Sy\implies x=y]$$ must be ...
2
votes
1answer
338 views

Model Theory-logic

Given the following formula: $$\bigg[\forall x P(x,x) \wedge \forall x \forall y \forall z\bigg( P(x,y) \wedge P(y,z) \Rightarrow P(x,z)\bigg) \wedge \forall x \forall y (P(x,y) \vee P(y,x)) \bigg] ...
6
votes
2answers
793 views

What is an example of a finite model in first order logic having a unique undefinable element?

This is (a slight paraphrase) of question 1.3.14 in Chang and Keisler's Model Theory book. "Show that for each natural number $n$, there is a language $L_n$ and finite model $M_n$ of $L$ such that ...
2
votes
1answer
151 views

Does this proposition hold if $\text{Mod}(\Gamma)=\emptyset$?

The following is the start of basic corollary in my logic text: For any set $\Gamma$ of sentences, $\Gamma\subseteq\text{Th}(\text{Mod}(\Gamma))$. What happens when $\text{Mod}(\Gamma)$ is empty? ...
15
votes
1answer
567 views

Is there a natural model of Peano Arithmetic where Goodstein's theorem fails?

Goodstein's Theorem is the statement that every Goodstein sequence eventually hits 0. It is known to be independent of Peano Arithemtic (PA), and in fact, was the first such purely number theoretic ...
10
votes
5answers
2k views

Example of non-isomorphic structures which are elementarily equivalent

I just started learning model theory on my own, and I was wondering if there are any interesting examples of two structures of a language L which are not isomorphic, but are elementarily equivalent ...