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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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 ...
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“Fraïssé limits” without amalgamation

All structures are countable with countable signature. Given a structure $\mathcal{A}$, the age of $\mathcal{A}$, $Age(\mathcal{A})$, is the set of structures isomorphic to finitely-generated ...
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Fields of characteristic zero via ultraproducts

Is every noncountable field of characteristic zero the ultraproduct (using a non principal ultrafilter over the set of prime numbers) of fields of positive characteristic?
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Can an ultraproduct be infinite countable?

In exercise 4 page 456 of Hodges "Model Theory" it is required to show that if an ultrafilter $\mathcal{U}$ is not $\omega_1$-complete, then every ultraproduct $\prod_I A_i/ \mathcal{U}$ has ...
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Stabilization of recursive approximation in $PA^-+I\Sigma_1^0$

Over any model M of $PA^-+I\Sigma_1^0$. Suppose $A\in [T]$ where $T$ is a $\Delta_2^0$-tree and $A$ is one isolated path. Further, $A$ is regular, i.e. $\forall n A\upharpoonright n$ has a code in ...
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Algebras admitting quantifier elimination

I apologize if this question is meaningless or trivial: What are examples of Algebras admitting quantifier elimination? Especially are there Groups admitting quantifier elimination? I need to say ...
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A preprint of Sela concerning the work of Kharlampovich-Miyasnikov

Yesterday, Z. Sela published a preprint in arXiv which claims that the solution of Olga Kharlampovich and Alexi Miyasnikov for the Tarski problem on decidablity of the first order theories of free ...
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Negated varieties and their relatively free algebras

During the past days, I asked some questions in order to gain a clear understanding of the notion of "free algebras". I suppose that the question below is the most clear image of the concept I have ...
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The existence of an algebra whose set of identities and first order theory are equivalent

Is there an algebra $A$ (for example a group) such that $Th(A)$ is logically equivalent to $id(A)$? In other words, is there an algebra $A$ such that $$ Mod(Th(A))=Var(A)? $$ Clearly finite algebras ...
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relatively free groups in $Var(S_3)$

Suppose $S_3$ is the symmetric group of order 6. Which elements of the variety $Var(S_3)$ are relatively free? This question is related to my previous question Relatively free algebras in a variety ...
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Universal graphs on higher cardinals

The Rado graph contains every finite graph as induced subgraph, and its also holds for countable graphs. So it is an universal graph of size $\aleph_0$, which contains all graphs of size $\aleph_0$ as ...
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Stability of analytic Zariski structures

Noetherian Zariski structures are introduced by Hrushovski and Zilber.(1996) An analytic Zariski structure is a generalization of Noetherian Zariski structures, introduced by Zilber and Peatfield. ...
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225 views

Is “approximate categoricity” absolute?

Let $T$ be a countable first-order theory, and assume that $T$ has exactly one atomic model up to isomorphism in every uncountable cardinality. (By "atomic" I mean a model which omits the ...
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Peano (Dedekind) categoricity

What is the smallest fragment of second order logic such that $Th(\mathbb{N})$ in that logic is categorical (only one model, namely natural numbers, up to isomorphism). For example, can we do this in ...
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239 views

Recursive ordinals and the minimal standard model of ZF

Does the minimal standard model of ZF contain all recursive ordinals or is it limited (probably by the proof theoretic ordinal of ZF as I suspect but cannot prove)? Paul J. Cohen's definition of the ...
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300 views

Recursive Non-standard Models of Modular Arithmetic? [closed]

Any algebraically closed field (ACF) is a model of Modular arithmetic (MA). (MA) has the same axioms as first order Peano arithmetic (PA) except $\forall x(Sx \neq 0)$ is replaced with $\exists ...
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Outer Definability of a Class

Definition: Let $C$ be a class of sets and $\mathcal{L}$ a first order relational language. We say $C$ is "outer definable" by $\mathcal{L}$ if there is a first order theory $T$ and for each $n_{R}$ - ...
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Is there a 0-1 law for the theory of groups?

Several months ago, Dominik asked the question Is there a 0-1 law for the theory of groups? on mathstackexchange, but although his question received attention there is still no answer. By asking the ...
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Generalizations of Birkhoff's HSP Theorem

Let $\mathbf{C}$ be the class of algebraic structures of some fixed type satisfying some sentence $\phi$. Birkhoff's HSP theorem says that $\mathbf{C}$ is closed under homomorphisms, subalgebras and ...
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$(\kappa,\lambda)$ - Minimal Models & Stronger Version of Rowbottom's Theorem

Definition 1: Let $M$ be a $\mathcal{L}$ - structure and $A\subseteq Dom(M)$. Define: $Def_{A}(M):=\{X\subseteq Dom(M)~|~\exists n\in \omega~~\exists \varphi (x,y_1,...,y_n)\in ...
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$(\kappa , \lambda)$ - Minimal Models of $\text{ZF}$

The notion of minimality in model theory is related to the existence of a gap in the size of definable subsets of a model. Now consider the following generalization: Definition 1: Let $M$ be a ...
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From elementary equivalence to isomorphism

A few years ago, when I took the basic course in Logic, I was very surprised to discover that given a signature $\sigma$ and two structures $M$ and $N$ of $\sigma$ which are elementarily equivalent ...
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Joint Forcing Extension Property

Definition 1: A class $\mathcal{K}$ of countable transitive models of $\text{ZF}$ has strong "joint forcing extension property" (JFEP) iff for all $M,N\in \mathcal{K}$ there are forcing notions ...
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Expressing “There are more Fs then Gs” in first-order logic

Is there a sentence of first-order logic that's true in all finite interpretations in which there are more Fs than Gs, and false in all finite interpretations in which there are not more Fs than Gs? ...
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What classes of complex manifolds are known to be definable in an o-minimal expansion of the real field?

It is a widely known (perhaps slightly folkloric) fact that compact complex manifolds, understood as first-order structures with a predicate for each analytic subset, are definable in an expansion of ...
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Cardinality of Grothendieck ring in model theory

Good evening, In model theory there is a notion of Grothendieck ring defined here http://math.berkeley.edu/~scanlon/papers/greu12jun00.pdf. Do we know anything about the cardinality of these rings ? ...
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Shelah's book on “Classification Theory”

As we know one of the most important and fundamental books in stability, simplicity, forking and ... classification theory, is Shelah's "Classification Theory" where lots of original ideas of the ...
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Turing Functional and $\Sigma_1^0$-formulas in models of fragments of PA

In models of PA with restricted induction power (for example, only $I\Sigma_n$ is present), the failure of higher induction scheme is characterised by the existence of definable cuts (like $\Sigma_2$ ...
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570 views

The interplay between certain aspects of interpretability, model theory and category theory

I have some questions about the interplay of interpretability, model theory and category theory. Since I had difficulties in finding literature or other helpful information about this topic, it would ...
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Vaught conjecture for uncountable languages

Recall Vaught conjecture: the number of countable models of a first-order complete theory in a countable language is finite or $\aleph_0$ or $2^{\aleph_0}.$ Now let $\lambda$ be an uncountable ...
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Is the consistency of $\mathcal{L}_{\infty\omega}$-sentences absolute?

The question is exactly that of the title. Suppose $\varphi\in V$ is an $\mathcal{L}_{\infty\omega}$-sentence, and $W$ is an inner model of $V$ such that $\varphi\in W$. Is the statement ...
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Recent application of model theory in set theory by Shelah-Malliaris

Recently one of the oldest open problems in set theory about the cardinal invariants of the continuum (i.e the question of whether $\mathfrak{p}=\mathfrak{t}$) was solved by Shelah and Malliaris (see ...
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Is there a Rado category?

The Rado graph appears to have a nice universality property (it contains all finite and all countably infinite graphs as induced subgraphs) and homogeinety property (any isomorphism between ...
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Sketches for categories of models of complete theories

In Accessible categories : the foundations of categorical model theory, chapter 3 p.58, Makkai and Paré claim that there is "an (obvious) identification of a class of sketches so that the categories ...
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Completeness theorem via syntactic categories

The nLab says in its internal logic article that the Completeness Theorem can be proven via a ``generic model'' of the theory. The model is generic in the sense that the only things true of it are ...
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Are descriptive and ontological notions of equality equal? [closed]

‎Let ‎$‎‎a$ ‎and ‎‎$‎‎b$ ‎are ‎two "‎objects". ‎What ‎is ‎the ‎meaning ‎of‎ ‎‎$a=b‎‎$‎? This is one of the deepest problems of philosophy and logic because one needs a complete information about ...
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Does existence of a proper class model imply the consistency?

The fundamental theorem of model theory says that: Theorem: A first order theory is consistent if and only if it has a model. In the above theorem we assume that the domain of any model is a ...
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Can we force with Fraisse filters to solve Vaught's conjecture?

Around the classic Fraisse amalgamation theorem in model theory we have the following notions: Definition (1): If $M$ be an $\mathcal{L}$-structure then define: $age(M):=\lbrace N~|~N~\text{is ...
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Is there a non-trivial consistency preserving transformation?

In ‎set ‎theory ‎"equiconsistency" (and not "consistency") ‎of ‎the ‎theories ‎is the‎ ‎main ‎part ‎of ‎researches. ‎So ‎we ‎usually ‎try ‎to ‎construct a‎ ‎new model ‎using a‎ ‎given ‎one. ‎In ‎the ...
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What are the Possible Large Cardinals of $L[X]$?

I've been doing some basic reading in inner model theory, and I'm at the point where I've seen the definition of things like Martin-Steel and Mitchell-Steel inner models. I am wondering about the ...
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Shelah's categoricity conjecture

Does Shelah's categoricity conjecture for abstract elementary classes have applications in other branches of mathematics?
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732 views

Where is the end of universe?

In some sense the empty set ($\emptyset$) and the global set of all sets ($G$) are the ends of the universe of mathematical objects. The world which $ZFC$ describes has an end from the bottom and is ...
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Absoluteness of completeness

Suppose $V_0, V_1$ are (not necessarily well-founded) models of ZFC and suppose $\varphi$ is a first order sentence in a finite language $L$ (in our background model of set theory). Because every true ...
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Is there a forcing closure?

The main theorem of forcing says that for any c.t.m of $ZFC$ like $M$ and for all partial order $\mathbb{P}$ and $\mathbb{P}$-generic $G$ over $M$, there is a c.t.m of $ZFC$, like $N$ such that $N$ is ...
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Are all complete finitely axiomatizable first order theories $\aleph_0$-categorical?

Suppose $T$ is a complete first order theory with a finite axiomatization. Must $T$ be $\aleph_0$-categorical? If not are there any simple examples of finitely axiomatized complete first order ...
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Relativization of Formulas and Models [closed]

I want to show that the definition of satisfiability is consistent with the definition given by relativization, i.e. Let $L=\{\in\}$. Let $M$ be a definable set and let $E\subset{M\times{M}}$. Let ...
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Do inner models of unique measurable cardinals have a regular behavior? (Edited and Revised Version)

We know that if ‎$‎‎\kappa‎‎$ is a‎ ‎measurable ‎cardinal ‎and ‎‎$‎‎\mu$ be a‎ ‎two-valued ‎non-trivial‎‎$‎‎‎\kappa‎$-additive ‎measure ‎on ‎it ‎the‎n the corresponding inner model produced by ...
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Is there an inner model between two distinct inner models of ZFC?

Definition (1): A‎n‎‎ ‎inner ‎model ‎of ‎‎$‎ZFC‎$ ‎is a‎ ‎tarnsitive proper class ‎model ‎‎of $‎‎ZFC$ ‎which ‎contains ‎all ‎ordinal numbers. ‎Informally ‎we ‎denote ‎the ‎collection ‎of ‎all ‎inner ...