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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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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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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380 views

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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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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178 views

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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147 views

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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1answer
145 views

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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218 views

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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236 views

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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60 views

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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221 views

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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226 views

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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114 views

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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377 views

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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488 views

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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344 views

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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104 views

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 ...
4
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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 ...
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Algebraically closed fields with only definable automorphisms

According to the paper "Models with second order properties IV. A general method and eliminating diamonds" by Shelah, he constructs ordered fields with only definable automorphisms. I haven't read ...
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Can the structure of an ultrafilter determine the structure of its ultrapower?

Usually we work with ultrafilters as pure sets without any structure. Q1. Is there any important notion of structure on an ultrafilter? Q2. Is there any non-trivial notion of structure on ...
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Is it ever a good idea to use Keisler-Shelah to show elementary equivalence?

The most useful way I know to show that two structures are elementarily equivalent is Ehrenfeucht-Fraisse games. These are quite nice and intuitive, and even when I can't use them to solve my problem ...
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Finite generation of vector identities

This question is partially motivated by Looking for a comprehensive referece for vector identities, although that question may not be appropriate for MO. Consider the set $\mathcal{E}$ of all valid ...
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A ZFC construction to get a proper extension which is a $\omega_1$-model

In $V$, let me call a set theory structure A is a $\omega_1$ model if the $\omega_1$ of $A$ is the same as the $\omega_1$ in $V$ (up to isomorphism). The question I would like to ask is the following: ...
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quantifier rank and number of variables as complexity measures

Is there a property of finite structures expressible with a sentence using only 2 variables and quantifier rank n but not expressible by any sentence with more variables and quantifier rank less than ...
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Varieties where every algebra is free

I'd like to know more about varieties (in the sense of universal algebra) where every algebra is free. Another way to state the condition is that the comparison functor from the Kleisli category to ...
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Vaught's conjecture for partial orders

In ``Steel, John R. On Vaught's conjecture. Cabal Seminar 76–77, pp. 193–208'' the following is proved: Theorem. Let $\phi\in L_{\omega_1,\omega}.$ If every model of $\phi$ is a tree, then $\phi$ has ...
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Can a Decidable Theory Have Non-recursive Models?

Tennenbaums' theorem proves neither addition nor multiplication can be recursive in any countable non-standard model of arithmetic. Tennenbaum's proof applies to theories much weaker than PA. ...
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Forcing for Arbitrary First Order Theories

Forcing is a relative model construction method for models of $ZF$ as a particular first order theory using models of another first order theory (forcing companion) that in this case is the theory of ...
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Godel's Second Incompleteness theorem and Models

As I understand it, Godel's completeness theorem essentially says that if a sentence $\phi$ can be proven in a first order theory $\Gamma$, then $\phi$ is satisfied in all models $\mathcal{U}$ of ...