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

learn more… | top users | synonyms

3
votes
1answer
141 views

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 ...
1
vote
3answers
293 views

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 ...
10
votes
3answers
857 views

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 ...
0
votes
0answers
22 views

CRAM[t(n) \subseteq IND[t(n)]

In the proof of lemma 5.3 in Immerman's descriptive complexity (CRAM[t(n) \subseteq IND[t(n)]), he mentions a disjunction depending on the value of the processor's program counter at the "directly" ...
4
votes
1answer
137 views

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 ...
5
votes
1answer
191 views

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: ...
2
votes
0answers
40 views

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 ...
12
votes
3answers
433 views

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 ...
8
votes
2answers
237 views

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 ...
2
votes
3answers
348 views

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. ...
4
votes
2answers
293 views

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 ...
3
votes
3answers
300 views

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 ...
6
votes
0answers
171 views

“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 ...
7
votes
1answer
239 views

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?
6
votes
1answer
288 views

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 ...
4
votes
0answers
85 views

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 ...
2
votes
0answers
142 views

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 ...
24
votes
1answer
2k views

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 ...
0
votes
3answers
133 views

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 ...
4
votes
3answers
249 views

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 ...
0
votes
1answer
149 views

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 ...
5
votes
2answers
222 views

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 ...
4
votes
0answers
194 views

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. ...
11
votes
1answer
228 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 ...
3
votes
0answers
108 views

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 ...
6
votes
1answer
241 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 ...
0
votes
1answer
304 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 ...
3
votes
1answer
138 views

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 $n_{R}$ - ...
49
votes
4answers
2k views

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 ...
6
votes
1answer
223 views

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 ...
2
votes
1answer
216 views

$(\kappa,\lambda)$ - 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 ...
1
vote
1answer
255 views

$(\kappa , \lambda)$ - 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 ...
2
votes
1answer
271 views

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 ...
3
votes
2answers
152 views

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 ...
2
votes
1answer
117 views

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? ...
6
votes
0answers
87 views

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 ...
4
votes
0answers
91 views

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 ? ...
8
votes
1answer
405 views

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 ...
3
votes
3answers
182 views

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 $\Sigma_2$ ...
12
votes
3answers
585 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 ...
6
votes
0answers
172 views

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 ...
6
votes
1answer
195 views

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 ...
5
votes
0answers
314 views

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 ...
7
votes
0answers
187 views

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 ...
4
votes
1answer
141 views

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 ...
3
votes
1answer
167 views

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 ...
3
votes
2answers
435 views

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 ...
8
votes
1answer
308 views

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 ...
4
votes
2answers
275 views

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 ...
1
vote
0answers
153 views

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 ...