# Questions tagged [model-theory]

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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### Is there difference in notion of measurability in classical versus constructive?

Are there notions of measure and examples of sets measurable in that measure in classical logic but not in constructive logic (I think there cannot be counterexamples in other direction)? Are there ...

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### Saturated differentially closed field

What means "saturated" in "saturated differentially closed field" ?

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### What is a good definition of a mathematical structure?

At the moment I am writing a textbook in Foundations of Mathematics for students and trying to give a precise definition of a mathematical structure, which is the principal notion of structuralist ...

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### Topological Vaught's conjecture for special theories

As is know, Vaught's conjecture is a special case of topological Vaught's conjecture.
On the other hand, the Vaught's conjecture is true for the following theories:
1- $\omega$-stable theories (...

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### Results of geometric model theory

I read in some places that model theory has useful applications in algebraic geometry.
Could someone give me some results that come from applying model theory to algebraic geometry?
I also saw that ...

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### Where is the flaw in this argument with $p$-adic extensions?

I cannot find what I am missing in the following computation. Let $K=\mathbb{Q}_p(p^{1/{(p-1)p^{\infty}}})$ and $L=\mathbb{Q}_p(\zeta_{p^{\infty}})$, where $\zeta_{p^n}$ is a primitive $p^n$-th root ...

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### Turing machines with all runs decidable

$\DeclareMathOperator\Comp{\mathit{Comp}}
\DeclareMathOperator\succ{\mathit{succ}}$Let $(\Phi_e)_{e\in\omega}$ be your favorite enumeration of Turing machines. For $e,n\in\omega$ there is a structure $...

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### Are Conway's combinatorial games the “monster model” of any familiar theory?

This is related to this question about a "mother of all" groups, and so seemed like it'd fit in better at MO than MSE.
If I understand the answer to that question correctly, the surreal numbers have ...

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### Tarski's truth theorem — semantic or syntactic?

I was reading the sketch of the proof of Tarski's theorem in Jech's "Set Theory", which appears as Theorem 12.7, thinking that it would be an interesting result to really understand. As stated in the ...

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### The lattice of analogues of Robinson's $Q$

This question was asked and bountied at MSE without response.
Call a sentence $\varphi$ in the language of arithmetic $Q$-like iff $\mathbb{N}\models\varphi$ and $\{\varphi\}$ is essentially ...

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### Definability in countable nonstandard models of Peano arithmetic

I know that every element of $\mathbb{N}$ is definable the standard model of Peano Arithmetic. Does there exist a countable non-standard model of PA where the same is true?

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### “Local” compactness properties beyond $\mathcal{L}_{\omega_1,\omega}$?

Below, all languages are finite for simplicity.
This question is about generalizations of Barwise compactness for logics more complicated than $\mathcal{L}_{\omega_1,\omega}$: properties of the form "...

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### Global choice for models of complete theories in $\mathsf{ZFC}$

This is a followup to a previous question of mine. To summarize the result of that question, by a result of Kanovei and Shelah, $\mathsf{ZFC}$ is enough to show that there is a uniform procedure for ...

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373 views

### Can nonstandard fields contain $\mathbb R$ in different ways?

Suppose $e : \mathbb R \to F$ is an elementary embedding in the language of ordered fields. Can there exist an elementary embedding $e' : \mathbb R \to F$ such that $e \not= e'$? Note that it would ...

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### Is there a characterization of the class of first-order formulas that are closed in every compact Hausdorff structure?

Fix a relational language $\mathcal{L}$. (I don't think relational really matters that much but I don't want to worry about it.) A topological $\mathcal{L}$-structure is an $\mathcal{L}$-structure $M$ ...

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255 views

### Cofinality of infinitesimals

Suppose $\kappa$ is an infinite cardinal and $U$ is a countably incomplete uniform ultrafilter over $\kappa$. Then $\mathbb R^\kappa/U$ is nonstandard. What is the cofinality of the set of ...

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### Where did this presentation of Godel's theorem appear?

This question was asked and bountied at MSE, with no response.
Many years ago I ran into the following proof of Godel's first incompleteness theorem
(here $T$ is an "appropriate" theory of ...

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112 views

### Preservation theorem for intersection of relations on a fixed set

It is a well known fact that any intersection of equivalence relations on a fix set is itself an equivalence relation. The same holds for a few other common relational theories, such as partial orders....

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### Is Thompson's group definably orderable?

Is Thompson's group $F$ definably left-orderable? definably bi-orderable?
Orderability definitions: Recall that a group $G$ is left-orderable (resp. bi-orderable) if it admits a left-invariant (resp. ...

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### Interpreting proper elementarily equivalent end extensions?

Is there a tuple of parameter-free formulas $\Phi$ and a nonstandard $M\models PA$ such that $\Phi^M\models PA$, the induced $M$-definable initial segment embedding $j_\Phi^M:M\rightarrow\Phi^M$ is ...

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201 views

### When are projective modules closed under highly-filtered colimits?

Let $R$ be a ring. Let $Mod(R)$ be the category of left $R$-modules, and let $Proj(R) \subseteq Mod(R)$ be the full subcategory of projective $R$-modules. Let's say that $R$-projectives are closed ...

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### Elementary equivalence between $n\mapsto n+1$ and its inverse on the Stone-Čech remainder?

Consider structures $(A,f)$ encoding a Boolean algebra $A$ endowed with an automorphism $f$. There is an obvious notion of isomorphism between such structures.
Consider the endomorphism $\hat{\Phi}$ ...

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### Is $\mathbb{F}_{p}(t)^{h}$ an elementary substructure of/existentially closed in $\mathbb{F}_{p}((t))$?

It is a well-known fact that the Henselization of the function field $\mathbb{F}_{p}(t)$ in regard to the $t$-adic valuation is $\mathbb{F}_{p}(t)^{alg} \cap \mathbb{F}_{p}((t))$, so of course $\...

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### What does second order set theory give us that is new?

There is a natural analogy between the theories PA and ZFC. See the linked question by Gro-Tsen here.
Peano arithmetic (PA) is a first order approximation to the natural numbers. As is well known, ...

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336 views

### The “contrary” of an isomorphism

Roughly, my question is: is there a standard name for functions which one might characterise as the "contrary" of an isomorphism?
Here is a more precise version of my question. Working model-...

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164 views

### Self-embeddings of uncountable total orders, 2

Let $S = (\Omega,\leq)$ be an uncountable dense total order, such that for all positive integers $m$ and all finite ordered sequences $a_1 < a_2 < \ldots < a_m$ and $b_1 < b_2 < \ldots &...

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144 views

### Improving a Lindstrom-y fact about $\mathcal{L}_{\omega_1,\omega}$?

See e.g. the last section of Ebbinghaus/Flum/Thomas for the relevant background on abstract model theory. Below, all languages are finite for simplicity. "$HC$" is the set of hereditarily countable ...

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149 views

### Elementary self-embeddings conservative over ZFC

Question: Is the following theory conservative over ZFC? And if not, what is its strength?
Language: $∈$, $j$ (unary function symbol)
Axioms:
1. ZFC (without separation and replacement for formulas ...

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### Category of finite models of a $\tau$-sentence

Let $\varphi$ be an $\tau$-sentence, we define the generalized spectrum of $\varphi$ as the class of its finite models, $$\text{GenSpec}(\varphi):=\{\mathcal{A}; \mathcal{A} \models \varphi, \lvert A\...

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### Do there exist acyclic simple groups of arbitrarily large cardinality?

Recall that a group $G$ is acyclic if its group homology vanishes: $H_\ast(G; \mathbb Z) = 0$. Equivalently, $G$ is acyclic iff the space $BG$ is acyclic, i.e. $\tilde H_\ast(BG;\mathbb Z) = 0$.
In ...

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### Ax theorem for separably closed fields

For the algebraically closed fields a theorem of Ax states that any injective polynomial map from $K^n$ to $K^n$ where $n\in \mathbb{N}$ and $K$ an algebraically closed field, is bijective.
Is there ...

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### Smallest ordinal modelling $\aleph_1$?

Let $X_1$ be the class of all ordinals $\alpha$ such that there exists a transitive model $M$ of ZF(C) such that $M$ thinks that $\alpha$ is $\aleph_1$.
Every class of ordinals has a minimum element (...

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### Equational theory in the signature (+,*,0,1) of sedenions and beyond

Consider a Cayley-Dickson algebra $(X,+,∗,0,1)$, that is an algebra generated from the reals by the Cayley-Dickson construction. From complexes to quaternions, we lose commutativity of multiplication, ...

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### Theorems of ZF through countable transitive models

Let $T$ be a finite collection of axioms of $\mathrm{ZF}$, let $\sigma$ be a sentence in the language of $\mathrm{ZF}$ and consider the statement
$\tau$: “any transitive countable model of $T$ ...

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### A Beth definability style result for imaginaries?

Let $T$ be a theory and let $I$ be an imaginary sort of $T$. We'll let $T_I$ denote the induced theory on $I$, by which I mean specifically the theory of all $\varnothing$-definable relations on $I$.
...

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### Model theory of the complex numbers with conjugation

Do there exist some results on the theory of $\mathbb{C}$ in the language $\{0,1+, \times, \overline{\cdot}\}$, where $\overline{\cdot}$ is the conjugation map $\overline{a+ib} = a - ib$?
I'm ...

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### Is there a procedure to derive models from axiomatic systems? [closed]

Is there a systematic procedure to construct a model of an axiomatic system from the system itself?
For example given the abstract postulates of a ring we can show that the integers satisfies them ...

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114 views

### Arithmetization of Syntax: Can any semantic be encoded as syntax?

It is my understanding that Gödel Encoding and "Arithmetization of Syntax" can be used to represent any logical system. This is exemplified by the encoding of a Universal Turing Machine.
"According ...

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360 views

### Can Godel's incompleteness theorems be in some sense circumvented this way?

New foundations "NF" (formulated in the language of $\small \sf FOL(\in)$), can define a kind of ordered pair relation $``\rho"$ such that we can have a set $E$ of those pairs where NF proves the ...

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### Quantifier elimination with no free variables and real polyhedral inequalities

In this introductory blog post https://cstheory.blogoverflow.com/2011/11/something-you-should-know-about-quantifier-elimination-part-i/ it is mentioned in the very last line that "I do not know if a ...

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### Lower bound for polyhedral real quantifier elimination

All known examples for double exponential lower bounds for real quantifier elimination involves polynomial inequalities with degree $>1$.
Is there an example of double exponentiality with ...

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### On certain order-automorphisms of the rationals

Consider the rationals $\mathbb{Q}$ with the usual order $\leq$. Now let $A$ be a subset of $\mathbb{Q}$, such that foreseen with the induced order $\leq$, $(A,\leq)$ is a dense linear order.
...

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### “Compactness for computability” - does it ever happen?

Throughout, "computable structure" means "first-order structure in a computable language with domain $\omega$ whose atomic diagram is computable."
Say that a computable structure $A$ is compact for ...

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### Fundamental theorem of linear orders

Let $(\Omega,\leq)$ be a countable linear order. Suppose that for every finite $m \in \mathbb{N}$, and all subsets $S_1$ and $S_2$ of $\Omega$ of order $m$, there is an order-automorphism of $(\Omega,\...

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### Compatible and incompatible sets [closed]

Definition of the compatibility relation
I have defined a relation $\mathsf{C}$ for sets that captures (to some extent) the notion of compatibility.
In order to do this, we need an operation $': \...

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915 views

### Is the union of a chain of elementary embeddings elementary?

I am a little confused about what I think must be either a standard theorem or a standard counterexample in model theory, and I am hoping that the MathOverflow model theorists can help sort me out ...

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### Is the equational theory of groups axiomatized by the associative law?

Consider the class of groups in the signature {*}. Is the equational theory of that class axiomatized by the associative law? I asked this on math stack exchange but I didn't receive a satisfactory ...

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### Survey article model theory research

I've taken a graduate course in model theory and I like it so much that I can imagine doing research in this area. Are there survey articles or review papers on the current research topics in model ...

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### On Asymptotic classes of finite structures (2)

As I mentioned in my previous question, I am reading the following paper:
One-dimensional asymptotic classes of finite structures.
by: Macpherson and Steinhorn
I have some crazy questions!
What is ...

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668 views

### Simple book on model theory

I was expressed by how Mendelson describes models in his Introduction to mathematical logic. Now I am looking for a nice model theory guide. The book (video source, etc.) must:
Include the concrete ...