Model theory is the study of (classes of) mathematical structures (e.g. groups, fields, graphs, universes of set theory) using tools from mathematical logic. Objects of study in model theory are models for formal languages which are structures that give meaning to the sentences of these formal ...

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non-axiomatizable logics

Hope you're all doing well. My question is about non-axiomatizable logics. My understanding is that a "logic" (the mathematical structure) is just another word for a "propositional calculus" as in ...
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64 views

Omitting Types… recursively

I'm working on the following problem at the moment: Let $\mathcal{L} = \{R\}$, where $R$ is a binary relation symbol. Let $T$ be a consistent, decidable $\cal{L}$-theory, and let $p(x)$ be a ...
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63 views

Complexity of index sets in nonprincipal ultrafilters

Let $U$ be a nonprincipal ultrafilter on $\omega$. It can be shown that the set $I = \{e \mid W_e \in U\}$ (where $W_e$ is the $e$th r.e. set in some given enumeration) cannot be $\Delta_2^0$ (in ...
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134 views

Standard models being non-standard?

If there is a ''set'' W in V which is a standard model of ZF, and the ordinal κ is the set of ordinals which occur in W, then Lκ is the L of W. If there is a set which is a standard model of ...
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generating complete consistent theories

This is a model theoretic question. I was reading Kremer & Mints's Dynamic topological logic paper, and it mentioned that by a “standard argument”, every consistent formula is a member of some ...
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First order logic, finite partially ordered structure problem

The exercise below is part of an exam for a Logic course. There were other questions leading up to this, but I've included their "result" in the wording below. Suppose alphabet $\mathcal{L} = ...
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115 views

Is this first order version of the Collatz conjecture decidable in peano arithmetic?

Let $\phi(x)$ be a first order formula in the language of arithmetic with one free variable $x$. Consider the sentence $\psi_\phi$, defined as: $$\phi(0)\wedge \phi(1) \wedge (\forall x \phi(x) \to ...
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202 views

Proving Tychonoff's theorem with the Compactness theorem of logic

It seems to be known that Tychonoff's Theorem for Hausdorff spaces and the Compactness theorem of first order logic are both equivalent over ZF to the ultrafilter lemma. Does anyone know a slick proof ...
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203 views

What's correspondence between the model theoric and the set theoric kernel of homomorphism?

A kernel of a mapping $h$ from $\mathfrak{A}$ to $\mathfrak{B}$, generally, is an equivalence relation $\{(a,a') \in \mathfrak{A} \times \mathfrak{A} \mid h(a)=h(a')\}$. However, in model theory, ...
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88 views

Automorphisms between two tuples of the same type

Fix some first order structure in some fixed language. If tuple $b$ is the image of tuple $a$ under some automorphism, then they have the same type. Is the converse true? That is, for any fixed tuples ...
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62 views

Can saturation drop when passing to an elementary extension.

Suppose I have a model $\mathcal{U}$ with some theory $T$ such that $\mathcal{U}$ is $\kappa$-saturated for some infinite cardinal $\kappa$. If $\mathcal{V} \succeq \mathcal{U}$ is an elementary ...
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358 views

A *finite* first order theory whose finite models are exactly the $\Bbb F_p$?

Since this question turned out to be trivial, I'm now asking this strengthened version: Is there a finitely axiomatized first order theory $T$ in the language of rings such that its finite models ...
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158 views

Question on the non-existence of a satisfaction formula in $\mathbb{L}$

I know this may sound trivial, but I'm having trouble figuring out what the problem is. Let $\mathbb{L}$ be the class of constructible sets. We know that $(\mathbb{L}_{\omega + \omega}, \in)$ is a ...
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145 views

A first order theory whose finite models are exactly the $\Bbb F_p$

Is there a first order theory $T$ in the language of rings such that its finite models are exactly the fields $\Bbb F_p$ with $p$ prime (but no $\Bbb F_q$ with $q$ a proper prime power is a model of ...
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206 views

first-order logic: Any finite set of first-order sentences true in all divisible abelian groups is true in some non-divisable one.

The question is the Subject. This comes from Barwise: Intro to First Order Logic, it is prop 2.1 and it is answered 2 pages later. I however don't understand the proof well. As Barwise says, the ...
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182 views

Why is there apparently no general notion of structure-homomorphism?

In model theory, one typically defines only embeddings of structures and isomorphisms, but I haven't seen a definition of general structure homomorphisms. Is there some particular reason behind that? ...
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Is there a universal property for the ultraproduct?

Given an ultrafilter U on a set I and a collection of ser X_i ($I \in I$) one defines the ultraproduct as the quotient of $\prod X_i $ by the identification $x_i=y_i :\leftrightarrow \{i:x_i=y_i\} \in ...
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What is exactly the meaning of being isomorphic?

I know that the concept of being isomorphic depends on the category we are working in. So specifically when we are building a theory, like when we define the natural numbers, or the real numbers, or ...
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103 views

Forking Independence

Marker proves in "Model Theory" that for an $\omega$-stable group, $G$, if it is connected then it has a unique generic type. I'm trying to understand the proof. The proof goes like this: Let $p,q$ ...
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552 views

Is Gödel's incompleteness theorem provable without any model-theoretic notion?

The entry on Gödel's incompletenss theorem in Wikipedia says: Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, ...
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92 views

Are there other, non-model theoretic, accounts of semantic validity?

When I first encountered the double turnstile ($\models$) in the context of an intro to logic class it was presented to me as expressing semantic validity where this was cashed out model-theoretically ...
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Henkin vs. “Full” Semantics for Second-order Logic and Multi-Sorted First Order Interpretations

In this paper by Jeff Ketland, he notes: With Henkin semantics, the Completeness, Compactness and Löwenheim-Skolem Theorems all hold, because Henkin structures can be re-interpreted as many-sorted ...
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91 views

Class of finite groups a Fraïssé Class? [duplicate]

Is the class of finite groups a Fraïssé class? Calling this class $K$, does $K$ satisfy the following: Joint embedding property Amalgamation property Hereditary property: if $G \in K$ and $H \le G$, ...
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55 views

Field extensions of $\prod \Bbb F_p /U$

The ultraproduct of all finite prime fields $ \Bbb F_p $ (over a nonprincipal ultrafilter U) is a field of characteristic 0. How do I show that it has exactly one extension of degree n for each ...
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292 views

Elementary embedding and elementary equivalence [duplicate]

I guess it is not difficult, but I spent an hour thinking about this without success. Elementary embedding implies elementary equivalence, but elementary equivalence between structures does not ...
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292 views

Is $\Bbb R$ definable in $(\Bbb C,0,1,+,*,\exp)$?

Is there a first-order formula ϕ(x) with exactly one free variable $x$ in the language of fields together with the unary function symbol $\exp$ such that in the standard interpretation of this ...
5
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1answer
273 views

finiteness and first order sentences

Lets consider a set of sentences $T$ and a signature $\sigma$. I proved (using compactness theorem) that when $T$ has arbitrary large models than also an infinite model. Now there are several ...
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328 views

Meaning of quote: “model theory = algebraic geometry - fields”?

On the wikipedia article for model theory, it says that a modern definition of model theory is "model theory = algebraic geometry - fields" and cites Hodges, Wilfrid (1997). A shorter model theory. ...
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512 views

Complete theories - dense linear order

There are two things I would like to prove. DLO - Dense linear order is complete, that means that when $\psi$ is a sentence of the language $\{<\}$ then $DLO\vDash\psi$ or $DLO\vDash\neg\psi$ ...
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159 views

Formalizing model-theoretical large cardinals in a formal system for ZFC

I work with the Metamath formal system, which is expressive enough to define ZFC and prove some nontrivial stuff, but my current goal is to define large cardinals, and I'm hitting a wall somewhere ...
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Two questions on stable groups

I'm going over Marker's Model Theory. I have two questions in the "$\omega$-Stable groups" section. In Lemma 7.1.12: $p\in S_1(G)$ and $\psi(v)$ defines $G^0$. He claims that there exists a $b\in G$ ...
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In ZF, does there exist an ordinal of provably uncountable cofinality?

Question is in the title. In ZFC, one can prove that $\aleph_{\alpha+1}$ is regular, so there is a large source of cardinals with uncountable cofinality, but in ZF, it is consistent that ${\rm ...
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Definability vs Automorphisms

(I am skipping any setup stuff and speaking roughly) One fact that I am sure of is that a definable subset $X$ is fixed by all automorphisms of the (super)structure. I simply wonder the converse: ...
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534 views

Is $ \pi $ definable in $(\Bbb R,0,1,+,×, <,\exp) $?

Is there a first-order formula $\phi(x) $ with exactly one free variable $ x $ in the language of ordered fields together with the unary function symbol $ \exp $ such that in the standard ...
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241 views

How is the standard model of number theory specified, and why can't we use that specification to prove any number theoretical sentence of interest?

According to Gödel's incompleteness theorem, there exists a sentence $G$ in the vocabulary of number theory ($N$) which is not provable from any (recursively enumerable) consistent set of axioms $T$, ...
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1answer
51 views

Existence of arbitrarily large ordinal subgroups in a group structure on a regular cardinal [duplicate]

Suppose $\kappa$ is an uncountable regular cardinal, and $(\kappa, \cdot, ^{-1}, e$) is a group. Prove that that $C = \{\alpha < \kappa: \alpha\, \textrm{is a subgroup of}\, \kappa)$ is unbounded ...
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Most astonishing applications of compactness theorem outside logic

The compactness theorem has a lot of applications to logic and model theory. I'm looking for applications. I'm looking for theorems in other areas of mathematics which seem at first sight to have ...
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1answer
54 views

Basics of Infinitary Formal Languages

Reading through Hodges' "A Shorter Model Theory", he gives the following symbolism (pgs. 23-25) for the first-order language constructed in the normal way with only finitely many formulas ...
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50 views

The set of primes as the spectrum of a first-order theory [duplicate]

In model theory, the finite spectrum of a first-order sentence $\phi $ (in a language with arbitrarily many constants, functions and relations is defined as the set of natural numbers $ n$ such that ...
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2answers
66 views

Constraint satisfaction problem - Arc consistency

The Constraint satisfaction problem (CSP) is roughly speaking a formalism that defines a finite set of relations over a domain. The relations are simply defined by enlisting elements in certain ...
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4answers
219 views

Models vs. Structures

Why are both the terms 'structure' and 'model' used in mathematical logic / model theory? Are they just holdovers from different subjects or is there a principled reason for having both? For ...
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4answers
170 views

Show that Total Orders does not have the finite model property

I am not sure whether my answer to this problem is correct. I would be grateful if anyone could correct my mistakes or help me to find the correct solutions. The problem: Show that Total Orders ...
2
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1answer
197 views

Different kinds of systems

I got interested in learning more about Logic, recently.The first thing i noticed is that this topic is a lot bigger than i expected. As i'm trying to make a sense of it all ( seeing the big picture ) ...
2
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1answer
125 views

Nonstandard structure of Presburger arithmetic

Let $\mathfrak {R}_A = (\Bbb {N}; 0, S,<,+)$. What can we say about ${}^{\ast}\Bbb N$, the universe of non-standard structure of the first order theory of $\mathfrak {R}_A$? Firstly, because of ...
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Online Model Theory Classes

Since "model theory" is kind of too general naming, I have encountered with lots of irrelevant results (like mathematical modelling etc.) when I searched for some videos on the special mathematical ...
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180 views

Does this logic have the downward Skolem-Löwenheim theorem?

Let $\mathcal L_Q$ denote the logic obtained from adding the quantifier $\newcommand{\almost}{\forall^\infty}\almost$ to the usual first-order logic, where the semantic interpretation of $\almost ...
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230 views

Does “=” have to be interpreted as equality?

To put it briefly: In model theory, we are allowed to interpret any relation symbol in any way we like. So why do people seem to require that "$=$" is interpreted as the actual equality? Let me ...
2
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2answers
279 views

How to show the relation $<$ is not definable in $(\Bbb N; 0, \operatorname {S})$ by quantifier elimination?

Show that the ordering relation $\{(m, n)| m < n \in \Bbb N\}$ is not definable in $\mathfrak{N}_{s}$. Suggestion: It suffices to show there is no quantifier-free definition of ordering. ...
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469 views

Axiom Systems and Formal Systems

I'm a really beginner in Mathematical Logic.I'm currently reading Shoenfield Mathematical's Logic and i'm having a hard time trying to relate the concept of Formal Systems with the concept of Axiom ...
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1answer
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Compactness principle via model theory.

A standard method of getting more concrete results from more abstract ones in Ramsey theory is the so called Compactness Principle. It is best illustrated by example. Here is the standard version of ...