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

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Groups and pregeometries

Definition. For an infinite structure $\mathcal{A}$ and $cl : P(dom(\mathcal{A})) \longrightarrow P(dom(\mathcal{A}))$ , we say that $(\mathcal{A}, cl)$ is a structure carrying an $\omega$-homogeneous ...
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Characterization of externally definable sets

Let $\cal U$ be a saturated model of inaccessible cardinality $\kappa$. For arbitrary $\cal D\subseteq U$ denote by $\langle\cal U,D\rangle$ the expansion of $\cal U$ with a new predicate for $\cal ...
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Does this property of a first-order structure imply categoricity?

Let $\mathfrak{A}$ be a first-order structure over a relational language and let $\kappa$ be an infinite cardinal. Lets say that $\mathfrak{A}$ has the $\kappa$-property if for every structure ...
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Are there superexponential Pfaffian functions?

This question is motivated by model theory, but it's really an analysis question (which means it may have an easy analysis answer that I just don't have the background for). Here's the main question, ...
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Can Suslin (or Aronszajn) lines ever be orderings of abelian groups?

I am interested in realizing linear orders as orderings of abelian groups. In particular, can Suslin lines (and other classes of line) be realised in this way? Let $\mathcal{C}$ be a class of ...
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dual (p,q)-property

If the set system $(X,S)$ has the $(p,q)$-property does its dual system also have the property? (Possibly, for different $p$ and $q$.) Explicitly, I am asking about the equivalence of the following ...
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Saturated Ultrapowers

I posted it on MSE and didn't receive any comments or answers, so I thought I would post it here. (See http://math.stackexchange.com/questions/895549/keisler-order-saturated-ultrapowers) Keisler's ...
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Representation of piecewise functions using variational inequalities

I need some help about how to represent piecewise smooth functions defined over N subintervals by using variational inequalities. From a paper I'm analyzing I get: "If we can find a function ...
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References for the Keisler Order

Are there any good/modern references on the Keisler order. I have been reading Keisler's original paper, "Ultraproducts which are not Saturated", which introduces the order. However it is somewhat ...
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definable relations in finite structures

Let $\mathcal{L}$ be a first order language and $M$ be a finite $\mathcal{L}$-structure. I want to know which relations $R\subseteq M^n$ are definable in $\mathcal{L}$? If I apply the well-known ...
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Is quasivariety generated by all perfect graphs finitely axiomatizable?

Fix logic $L$ with equality and a binary relation symbol $E$. The class of graphs can be identified with the class of models of the universal first-order Horn $L$-sentences $\forall x,y\; E(x,y) ...
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Countable model theory for $\omega$-stable theories?

This is a bit of a fishing expedition, because I'm not sure what I'm looking for. Very vaguely stated, here's the driving question: What conditions on an $\omega$-stable theory make the class of ...
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255 views

How do we know if Vaught's Conjecture is Absolute?

Please note that this might be some confusion on my part about the work surrounding Vaught's conjecture. First of all, Vaught Conjecture states that if a first-order complete theory $T$ in a ...
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Main Gap Phenomenon

Shelah's Main Gap Theorem states that for all first-order, complete theories, T, in a countable language, we have that either $$I(T,\aleph_\alpha)=2^{\aleph_\alpha}$$ or ...
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472 views

A double centralizing theorem for finite groups

I have a proof for the following assertion which employs Model Theory. It has certainly a pure group theoretic proof, but what is such a proof? Is the assertion trivial? Theorem Let $G$ be a finite ...
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Elementary chains of $\aleph_1$-saturated models

If $X$ and $Y$ are two sets linearily ordered by $<$, $X$ is called cofinal in $Y$ if $X \subseteq Y$ and and for every $y \in Y$, there is a $x \in X$ with $y < x$. If $M$ is some model and ...
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What axioms (other than choice) have a taming effect on the ordering of cardinalities?

Axiom of choice arranges all cardinalities into a well-ordered chain but without it their ordering can be wild in general ZF models, e.g. two cardinalities may not even have inf or sup. However, ...
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How to prove a Lefschetzesque principle for relative homology and absolute homology

Here is my conjecture: If H is some homology theory from some abelian category A to some abelian category B, E is the class of all epimorphism in A, E' is some closed class of epimorphisms in A and ...
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The number of countable models [duplicate]

Let $\mathcal{L}$ be a countable first order language. For a natural number n, can we find a complete $\mathcal{L}$-Theory $T$ which has exactly n non-isomorphic countable models ?
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Isomorphism of real closed fields

Given two real closed fields $R_1$ and $R_2$ such that both have cardinality continuum, archimedean, but not necessarily complete. Assume further that they are back and forth equivalent (in the ...
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A “suitably generic” set of Cohen reals without forcing?

I was reading a paper by David Marker, whose main theorem was that if $T$ is a first-order theory which is not small, then $F_2\leq_B \cong_T$. That's not especially relevant to the question at hand, ...
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Models of PRA/EFA with induction on $X$ but not $\omega^X$

As I currently understand it, induction on formulas containing $N+1$ first-order quantifiers is required to prove the well-ordering of the ordinal $(\omega \uparrow\uparrow N) < \epsilon_0$, that ...
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Is the following a sufficient condition for being a primal algebra?

I have a question regarding universal algebra and, in particular, primal algebras: Suppose that A is a finite simple algebra with no proper subalgebra, no automorphism except the identity map, with a ...
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Counterexample for closedness under union of $\prec_{\infty,\kappa}$ chains

Assume $\kappa$ is uncountable and $\phi$ is an $L_{\infty,\kappa}$ sentence. Let $K$ be the collection of models of $\phi$ partially ordered by $\prec_{\infty,\kappa}$. It is well-known that $K$ is ...
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A model with $\kappa$ many automorphism and a rigid element.

The following should be known, but I could not find an example. Let $\kappa$ be an uncountable cardinal. Find a model $M$ of size $\kappa$ which has $\ge\kappa$ many automorphisms, but for some $m\in ...
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How can one define the direct limit of classes?

If we have a family of classes $(\mathfrak{M}_\alpha)_{\alpha\in D}$ of $\in$-structures with $D$ being a limit ordinal or the class of ordinals, and a family ...
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Interactions between (set theory, model theory) and (algebraic geometry, algebraic number theory ,…)

Set theory and model theory have many applications outside of logic, in particular in algebra, topology, analysis, ... On the other hand model theory, in particular after Hrushovski, found many ...
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Overspill in models of arithmetic

Assume that $M$ is a non-standard model of complete arithmetic, i.e. of the theory $Th(\mathbb{N})$. Suppose that $R$ and $S$ are proper cuts of $M$. (With a cut, I mean a subset of the universe of ...
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Scott sentence in models of set theory

Let $\mathfrak{M}$ be a countable transitive model of set theory. Let $L$ be some countable language and $A$ be a countable (in $\mathfrak{M}$) $L$-structure. My question is: In $\mathfrak{M}$ can ...
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Foundation scheme for $\Sigma_{n+1}$-formulas

I have trouble working out a proof in the second part of Jean-Pierre Ressayre and Alex Wilkie. Modèles non standard en arithmétique et théorie des ensembles. Publications ...
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Is this dichotomy for a VC-shatter-like function known?

Associated with each family $\mathcal{F}$ of a subsets of a set $\mathbf{X}$ I have a function $f:\mathbb{N}\rightarrow\mathbb{N}$ resembling the dual shatter function. This function obeys a ...
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Sufficient Condition for Defining $\in$

Consider the first order language $\mathcal{L}=\{\in,\in'\}$ with two binary relational symbols $\in , \in'$ and $ZFC$ as a $\{\in\}$-theory. If we define $\in'$ using $\{\in\}$-formula $\varphi(x,y)$ ...
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A Special Pair of Formulas

Consider the first order language ‎$‎‎‎\mathcal{L}=\{\in,\subseteq\}‎$ and ‎$‎‎\{\in\}$-theory ‎$\text{ZFC}$.‎ ‎Is ‎the‎re a formula ‎$‎‎\psi ‎(x,y)‎ \in \{\subseteq\}-Form‎$ ‎with ‎the ‎following ...
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infinitary logic and partial fixed point logic

Is there a property definable in finite-variable infinitary logic $L^{\omega}_{\omega\infty}$ but not definable in partial fixed point logic FO(PFP) ?
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Quantifier elimination vs decidability

Quantifier elimination is used as a technique to get decidability (e.g. $Th( \mathbb{N}, +)$ ) of theories, but typically one has to go over to some expansion. Are there examples of theories which are ...
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Systematic brute-force searches for counterexamples

This is getting nowhere on math.stackexchange.com, so I'm putting it here. Gödel's completeness theorem says that for every statement in first-order predicate calculus with equality, there is either ...
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What is the precise notion of “enough arithmetic” in Godel's first Incompleteness theorem?

I'm trying to reconstruct the proof of Godel's first theorem (Rosser's strong version) from the uncomputability of the Halting function. If we just started with the language $\mathcal{L}=\{0, S, +, ...
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The (non-)absoluteness of second-order elementary equivalence

Elementary equivalence is set-theoretically absolute between any two transitive models of set theory; this is also true for the infinitary logics - e.g., $\mathcal{L}_{\omega_1\omega}$ - at least, ...
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Preserving Predimension Functions under Functional Convergences

Definition 1. If ‎$‎‎‎\mathcal{L}‎$ ‎is a‎ ‎countable relational ‎language, ‎a ‎predimension ‎class ‎‎‎‎‎$‎C‎$ is a class ‎of $‎‎\mathcal{L}$-structures with ‎the ‎following ‎properties:‎ ‎C1: ...
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Preimages of accessible full subcategories

My question is ultimately about the model theory of $L_{\infty, \omega}$, but it is more convenient to phrase it in terms of category theory. Suppose I have finitely accessible categories ...
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Stable examples from Algebra such that the model theoretic algebraic closure of a substructre is no model

Let $T$ be a stable theory. Let $A$ be a subset or substructure of a model $M$ of $T$. Now in some theories the (model theoretic) algebraic closure of $A$ is already a (sub)model of $T$. For example, ...
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Relation between fundamental theorem of Ehrenfeucht-Fraisse games and notions of bisimulation and simulation

Vijay D alluded to the relation between the fundamental theorem of Ehrenfeucht-Fraisse games and notions of bisimulation and simulation in response to the whats-a-magical-theorem-in-logic question. ...
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Is a model of arithmetic contained in a model of arithmetic an initial segment?

It's easy enough to show that if $\mathbb{N}_1$ is a non-standard model of the Peano axioms, then there is a canonical embedding $\mathbb{N} \to \mathbb{N}_1$, and we have a theorem that if $x \in ...
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Maximality of linear orders in the Keisler order on theories

Recently Malliaris and Shelah (see their preprint http://math.uchicago.edu/~mem/Malliaris-Shelah-CST.pdf) have shown that theories with $SOP_2$ are maximal in the Keisler order. A preceding result of ...
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Compactness and completeness in Gödel logic

The standard proof of the completeness theorem in first-order Gödel logic is based on a first-order countable language. I want to know that is there any proof of the completeness theorem in ...
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Equational theories determined by “identities without variables”

How to characterize equational theories $T$ which have the following property: for any two terms $t(x_1,...,x_n)$ and $t'(x_1,...,x_n)$ in the signature of $T$, if for any closed terms (i. e. terms ...
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unique types and decidability

Suppose $\mathcal{M}$ is an infinite structure which has the property that every type that is realised is realised uniquely. Also assume that every element of $\mathcal{M}$ is definable but there is ...
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Colorful model theory

There are a number of concepts in model theory - often situated around Hrushovski's amalgamation method (see for instance http://math.univ-lyon1.fr/~wagner/nijmegen.pdf) - which are colorfully named: ...
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Morley Phenomena for Special Families of Reals

Vaught's Conjecture is a dual form of Continuum Hypothesis in model theory. It asserts that for each complete consistent theory $T$ in a countable language if $I(T,\aleph_{0})>\aleph_{0}$ then ...
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Replace Morley sequence over some set by one over a finite set, s.t. they both satiesfy a certain formula

Let $T$ be a stable $L$-theory with elimination of imaginaries. We work in the monster model $\mathfrak C$ of $T$. Let $A$ be a small (infinite) set of the monster, $\phi(x,y)$ be a $L(A)$-formula and ...