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 ...

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347 views

Non-axiomatisability and ultraproducts

Let $T$ be a first-order theory over a language $L$, and let $\mathcal{M}$ be a subclass of the class of models of $T$. As I understand it, if there is no theory $\hat{T}$ over $L$ whose class of ...
10
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244 views

Number of automorphisms of saturated models

I have the following assignment question: Let $M$ be an $L$-model of cardinality $\kappa$. Assume $M$ is saturated. How can you show that $|\text{Aut}(M)|=2^{|M|}$? I see two possible ...
8
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160 views

Functoriality of the correspondence between oligomorphic actions and $\aleph_0$-categorical theories

If a group $G$ acts on a set $X$, then the action is said to be oligomorphic if the number of orbits of $X^n$ under the action is finite for each $n$. There is a classic theorem in model theory that ...
6
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102 views

Non-standard proofs of standard theorems

In Richard Kaye's book Models of Peano arithmetic, one can read (page 13): We have proved that any nonstandard $M \models \mathrm{Th}(\mathbb{N})$ has a nonstandard $a \in M \models \theta(a)$ iff ...
6
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251 views

Using the compactness theorem to show a set of first-order formula is equivalent to a set of quantifier-free formula

I am going through some theorems in Hodges' ``A shorter model theory'' and I have realized that I do not understand a certain argument regarding compactness. My question has two forms, I am sure that ...
5
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87 views

complete, finitely axiomatizable, theory with 3 countable models

Does it exist a complete, finitely axiomatizable, first-order theory $T$ with exactly 3 countable non-isomorphic models? A few relevant comments: There is a classical example of a complete theory ...
5
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85 views

ind-completion and functors which are full with respect to isomorphisms

Let $C$ and $D$ be categories and $F:C\rightarrow D$ a faithful functor which is full with respect to isomorphisms. This means that if $a,b\in C$ and $f:F(a)\rightarrow F(b)$ is an isomorphism in $D$, ...
5
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131 views

Gödel's Completeness Theorem and logical consequence

At the end of a long process of "rumination" on "old" math log textbooks, I've found the "missing link" - from my personal point of view - between some issues I've raised in the previous months : (i) ...
5
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0answers
108 views

Model theory in terms of type spaces/Lindenbaum algebras

Are there any good references that go into some detail of known 'translations' between properties of the type space of a model and the model theoretic properties of the model? All I seem to find are ...
5
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318 views

Structuralist slogans

I am afraid to make a bad impression by misusing this forum but I am looking for as-many-as-possible mathematically inspired formulations and references to one (sometimes vague) idea. The idea is ...
4
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67 views

Models of infinite groups and 'Group-like' objects

Let $G$ be an infinite group, and for simplicity, we will assume that $G$ is also countable. Now, with $G$ in mind, we construct a new language $L_G=\{f_{a_i\_},f_{\_a_i}:a_i\in G\}$ where ...
4
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48 views

Categorical description of permutation-invariance of models

One of my projects (which I should really leave to professionals, but I'm trying anyway) is trying to find a categorical description of the permutation-invariance of models of NF, if there is such a ...
4
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197 views

Characterization of the First Order Theory of Ordered Abelian Groups via Quantifier Elimination

In the paper "Elimination of Quantifiers in Algebraic Structures" Macintyre, McKenna and van den Dries, proved that every field (ordered field) whose theory admits quantifier elimination in the ...
4
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193 views

Difference between elementary submodel and elementary substructure

This is a really "elementary" question, forgive the pun. What is the difference between an elementary submodel and an elementary substructure (in first-order Logic)? Sincere thanks for help.
4
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173 views

What is the correspondence between the semantic compactness of logics and the compactness of Stone spaces?

According to the answer by François G. Dorais, we know that a logic $\mathfrak L$ is compact iff its Stone space of the Lindenbaum–Tarski algebra of the empty theory (w.r.t the deductive system) is ...
4
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109 views

is a group that eliminates quantifiers in the group language Abelian?

Let $G$ be a group. Consider it as a structure in the language of groups $L=(e,\cdot, (-)^{-1})$. Suppose that $G$ eliminates quantifiers in this language. Is it true that $G$ must be Abelian then? ...
3
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26 views

Let $h: A \to B$ be a weak homomorphism. Is h$[A]$ a substructure of $B$?

A little bit more precise: let $\mathfrak{A}$ and $\mathfrak{B}$ be two structures. Define a weak homomorphism as a function $h: \mathfrak{A} \to \mathfrak{B}$ such that the folowing conditions are ...
3
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85 views

Countable transitive $T \vDash ZFC-P$ with $\approx$ not absolute: why do we need $H(\aleph_3)$?

I am trying to solve Exercise IV.3.31 from Kunen's Foundations of Mathematics. I think I have a solution but I am confused by one of the hints. By request, here is the text of the exercise. ...
3
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0answers
74 views

Examples of Jónsson Models

Let $T$ be a complete first order theory. Suppose that $M\models T$. Then, $M$ is said to be a Jónsson Model of $T$ if for all $N$, such that $N\prec M$ and $N\models T$, we have $|N|<|M|$ (Note ...
3
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70 views

Finding a finite model

Hello I am having difficulty with this question, I am not even sure what strategy one would go about proving something like this: Suppose $L$ is a language which includes an infinite list ...
3
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63 views

Question on a Theorem from Chang-Keisler's Model Theory concerning $\Sigma^0_n$ sentences

The Theorem is 3.1.11 and states that for $n>0$ the following are equivalent : $\phi$ is equivalent both to a $\Sigma^0_{n+1}$ and a $\Pi^0_{n+1}$ sentence. $\phi$ is equivalent to a Boolean ...
3
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138 views

Crankery: Is there a perfect inner model of ZFC?

In the book "Aspects of Vagueness", the article "The alternative set theory and its approach to Cantor's set theory" by A. Sochor proposes the following definition: We will say that a set universe ...
3
votes
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87 views

generating complete consistent theories

This is a model theoretic question. I was reading Kremer & Mints's Dynamic topological logic paper, and it mentioned that by a “standard argument”, every consistent formula is a member of some ...
3
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125 views

Formalizing model-theoretical large cardinals in a formal system for ZFC

I work with the Metamath formal system, which is expressive enough to define ZFC and prove some nontrivial stuff, but my current goal is to define large cardinals, and I'm hitting a wall somewhere ...
3
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93 views

Online Model Theory Classes

Since "model theory" is kind of too general naming, I have encountered with lots of irrelevant results (like mathematical modelling etc.) when I searched for some videos on the special mathematical ...
3
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0answers
99 views

The Logic of Satisfiability?

I am aware of some study into the logic of provability. It is generally taken to be intermediate in strength between S4 and S5 modal logics. Is there corresponding study into something like the logic ...
3
votes
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164 views

A question about a passage in Just/Weese's Basic Set Theory

I have a question regarding the following passage in Just/Weese (p 192), $\mathbf{V}$ denoting the cumulative hierarchy: "But consider the following situation: Where in "($\beta$)" does the ...
3
votes
0answers
114 views

Some intuition behind o-minimal systems.

I am following the definitions of an o-minimal system as defined in "Tame Topology and O-minimal Structures" by Lou van den Dries. It is immediate from the definition that the graph of $\sin(x)$ is ...
3
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81 views

An application of Descriptive set theory in Model theory.

In page 162 of D.Marker Model theory book he proved that the set $S_n(F,T)$ of all $F$-types realized by some $n$-tuple in some countable model of $T$ is analytic (this is with any $F$:= $countable$ ...
2
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0answers
49 views

Prove that $\Pi_{i \in I} \mathbf{A}_i \cong \Pi_{j \in J} (\Pi_{i \in I_j} \mathbf{A}_i)$ where $\{I_j \ | \ j \in J\}$ is a partition of I.

My problem is following: prove that $\Pi_{i \in I} \mathbf{A}_i \cong \Pi_{j \in J} (\Pi_{i \in I_j} \mathbf{A}_i)$, where $\langle \mathbf{A}_i \ | \ i \in I \rangle$ is an indexed set of similar ...
2
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0answers
49 views

On the back and forth conditions for a set of partial isomorphisms

I've recently begun reading Poizat's A Course in Model Theory and already in the first pages I had some doubts. One odd (not necessarily bad) thing is that he defines notions such as isomorphism only ...
2
votes
0answers
106 views

How large or small can the gap in Shelah's main gap theorem be, up to consistency?

Shelah's main gap theorem in model theory says: For each first order complete theory $T$ in a countable language if $I(T,\kappa)$ denotes the number of its models of size $\kappa$ then one of ...
2
votes
0answers
51 views

The composition of functors and algebraic structures

An algebraic structure can be viewed as a finite-product-preserving functor $X : \mathbf{L} \rightarrow \mathbf{C},$ where $\mathbf{L}$ is a Lawvere theory and $\mathbf{C}$ is a category. Thus given ...
2
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85 views

Semantic Proof of Tarski's Undefinability of Arithmetic Truth

A few years ago I took a logic course and I've since lost my notes. I seem to remember a very semantic proof of Tarski's theorem on the undefinability of arithmetic truth, one that didn't use the ...
2
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38 views

Books/papers on model theory in non-monotonic logic

I am working on a project whose object language is in non-monotonic logic. Since the project involves reasoning about the models, I am thinking of translating a non-monotonic problem into a ...
2
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0answers
100 views

Isomorphism of finite models

Let $\mathfrak A$ and $\mathfrak B$ are models of finite signature $\sigma$. Prove that $\mathfrak A$ and $\mathfrak B$ are isomorphic, if $\mathfrak A \equiv \mathfrak B$ and $\mathfrak A$ is ...
2
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74 views

Lowering the power of infinite model

I need to prove that for every infinite model $\mathfrak A$ of signature $\sigma$ exists model $\mathfrak B$ with attributes: $\mathfrak A \equiv \mathfrak B$. $\parallel \mathfrak B \parallel = ...
2
votes
0answers
112 views

Recursive non-standard models?

Any algebraically closed field (ACF) is a model of Modular Arithmetic (MA). MA has the same axioms as first order Peano arithmetic except $\forall x(Sx \neq 0)$ is replaced with $\exists x(Sx=0)$. MA ...
2
votes
0answers
38 views

Ways to build models with specific properties

I'm studying Model Theory: an introduction by David Marker and more specifically doing the exercises of chapter 2 ("Basic techniques"). In several of these exercises, one is asked to build new models ...
2
votes
0answers
38 views

Given a Hardy Field is it always possible to find a smooth representative of each germ?

In this case I refer to a Hardy Field (of germs at infinity) $\mathcal{H}$ a a field of germs of real valued functions on $\mathbb{R}$ that is closed under differentiation. That is, if ...
2
votes
0answers
131 views

Defining “structured sets”

In his Notes on Set Theory (p. 44) Moschovakis defines: A structured set is a pair $U = (A,S)$ where $A$ is a set, the space of $U$, and $S$ is an arbitrary object, the frame of $U$. But even ...
2
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0answers
51 views

Countable Ultrahomogeneous Structures

I've been learning about countable ultrahomogeneous structures, where ultrahomogeneous means every isomorphism of finitely generated substructures extends to an automorphism of the whole structure. ...
2
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0answers
41 views

What are some ways of showing that a structure is not minimal?

The question is really in the title. My background in model theory is very limited. Basically nothing past the definition of minimal structures and minimal subsets. I am interested in some ...
2
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86 views

What kind of logics satisfy the coincidence lemma?

Lets formulate the incidence lemma as follows. We have a possibly infinite set of variables X and the domain of discourse U. Lets define an interpretation of the variables X in the domain U as a ...
2
votes
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43 views

Concurrent relation and enlargement

The superstructure $V({}^{\ast}X)$ [with respect to a monomophism $\ast : V(X) \to V({}^{\ast}X)$] is called an enlargement of $V(X) $ if for each set $A \in V(X)$ there is a set $B \in {}^{\ast} ...
2
votes
0answers
50 views

Model complete theories of henselian local rings which are not nec valuation rings

I just want to ask if anybody as any examples of a first order model complete theorie of henselian local rings which is not some theory of valuation rings. More precisely- I am looking for a theory ...
2
votes
0answers
114 views

Realizing n-types of a complete theory

Let $\mathfrak{A}_0=(A_0,R_0)$ and $\mathfrak{A}_1=(A_1,R_1)$ be structures such that for each $i<2$, $A_i$ is a nonempty set and $R_i\subseteq A_i\times A_i$. We define the structure ...
2
votes
0answers
145 views

Can we find a nice definition of Congruence in Topology?

According to my knowledge, quotient structure is a original structure divided by a congruence. However, quotient topology space is defined this way. Quotient_topology In this way, $\sim$ is only said ...
2
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0answers
51 views

Nice model theoretic properties of a theory adding a new predicate?

I would like to know what nice model theoretic properties (for example simplicity, NIP, stability, etc) can be preserved when we add a new predicate to the language. Explicitly, if $T$ is an L-theory ...
2
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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 ...