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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373 views

Non-axiomatisability and ultraproducts

Let $T$ be a first-order theory over a language $L$, and let $\mathcal{M}$ be a subclass of the class of models of $T$. As I understand it, if there is no theory $\hat{T}$ over $L$ whose class of ...
8
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170 views

Functoriality of the correspondence between oligomorphic actions and $\aleph_0$-categorical theories

If a group $G$ acts on a set $X$, then the action is said to be oligomorphic if the number of orbits of $X^n$ under the action is finite for each $n$. There is a classic theorem in model theory that ...
7
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118 views

Non-standard proofs of standard theorems

In Richard Kaye's book Models of Peano arithmetic, one can read (page 13): We have proved that any nonstandard $M \models \mathrm{Th}(\mathbb{N})$ has a nonstandard $a \in M \models \theta(a)$ iff ...
7
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272 views

Using the compactness theorem to show a set of first-order formula is equivalent to a set of quantifier-free formula

I am going through some theorems in Hodges' ``A shorter model theory'' and I have realized that I do not understand a certain argument regarding compactness. My question has two forms, I am sure that ...
6
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115 views

complete, finitely axiomatizable, theory with 3 countable models

Does it exist a complete, finitely axiomatizable, first-order theory $T$ with exactly 3 countable non-isomorphic models? A few relevant comments: There is a classical example of a complete theory ...
6
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183 views

Gödel's Completeness Theorem and logical consequence

At the end of a long process of "rumination" on "old" math log textbooks, I've found the "missing link" - from my personal point of view - between some issues I've raised in the previous months : (i) ...
6
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208 views

Characterization of the First Order Theory of Ordered Abelian Groups via Quantifier Elimination

In the paper "Elimination of Quantifiers in Algebraic Structures" Macintyre, McKenna and van den Dries, proved that every field (ordered field) whose theory admits quantifier elimination in the ...
5
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48 views

Type-definable Forcing or forcing in a non-first order setting

Roughly speaking, in set forcing the forcing notion is a set from ground model's perspective and in class forcing its a definable subset of the ground model given by solutions of some formula with ...
5
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56 views

Axiomatizability of the multiplication of a ring

The operation of ring multiplication is axiomatizable, if we allow ourselves an additional auxillary addition symbol. Just write down the ring axioms in the signature $\{*,+\}$. But could ...
5
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117 views

ind-completion and functors which are full with respect to isomorphisms

Let $C$ and $D$ be categories and $F:C\rightarrow D$ a faithful functor which is full with respect to isomorphisms. This means that if $a,b\in C$ and $f:F(a)\rightarrow F(b)$ is an isomorphism in $D$, ...
5
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112 views

Model theory in terms of type spaces/Lindenbaum algebras

Are there any good references that go into some detail of known 'translations' between properties of the type space of a model and the model theoretic properties of the model? All I seem to find are ...
5
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327 views

Structuralist slogans

I am afraid to make a bad impression by misusing this forum but I am looking for as-many-as-possible mathematically inspired formulations and references to one (sometimes vague) idea. The idea is ...
4
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72 views

Models of infinite groups and 'Group-like' objects

Let $G$ be an infinite group, and for simplicity, we will assume that $G$ is also countable. Now, with $G$ in mind, we construct a new language $L_G=\{f_{a_i\_},f_{\_a_i}:a_i\in G\}$ where ...
4
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0answers
51 views

Categorical description of permutation-invariance of models

One of my projects (which I should really leave to professionals, but I'm trying anyway) is trying to find a categorical description of the permutation-invariance of models of NF, if there is such a ...
4
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143 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 ...
4
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183 views

What is the correspondence between the semantic compactness of logics and the compactness of Stone spaces?

According to the answer by François G. Dorais, we know that a logic $\mathfrak L$ is compact iff its Stone space of the Lindenbaum–Tarski algebra of the empty theory (w.r.t the deductive system) is ...
4
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121 views

Some intuition behind o-minimal systems.

I am following the definitions of an o-minimal system as defined in "Tame Topology and O-minimal Structures" by Lou van den Dries. It is immediate from the definition that the graph of $\sin(x)$ is ...
4
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112 views

is a group that eliminates quantifiers in the group language Abelian?

Let $G$ be a group. Consider it as a structure in the language of groups $L=(e,\cdot, (-)^{-1})$. Suppose that $G$ eliminates quantifiers in this language. Is it true that $G$ must be Abelian then? ...
3
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73 views

Motivation behind automorphism bases?

Given a model $\mathcal{M}$ with a domain $M$ and $B \subseteq M$, $B$ is an automorphism base for $\mathcal{M}$ iff $\forall f \in Aut(\mathcal{M}). (\forall b \in B. f(b)=b) \implies f = ...
3
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65 views

Consequence in $\mathcal{L}_{\infty\lambda}$

Consider the infinitary first-order language $\mathcal{L}_{\infty\lambda^+}$ whose non-logical vocabulary consists of $\lambda \geq \omega$ individual constants and countably many predicate constants ...
3
votes
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46 views

Ordering of $\mathbb{R}$ not quantifier-free definable in $L_{R}$

I'm reading David Marker's book "Model Theory: An Introduction" and I'm trying to solve Exercise 3.4.24 which is stated as follows: Let $x$ and $y$ be algebraically independent over $\mathbb{R}$. a) ...
3
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38 views

Uncountably Categorical Theories and Embeddings

Suppose that $T$ is uncountably categorical. By the Baldwin-Lachlan Theorem, we note that $I(T,\aleph_0)=1$ or $\aleph_0$. Suppose that $I(T,\aleph_0)=\aleph_0$. Is it always the case that we get ...
3
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35 views

An affine group behaving like a field

This question is about an example of interpreting a field in an affine group, from Section 1.3 of Marker's Model theory: An introduction. Let $F$ be an infinite field and $G$ be the group of ...
3
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88 views

Prove that $\Pi_{i \in I} \mathbf{A}_i \cong \Pi_{j \in J} (\Pi_{i \in I_j} \mathbf{A}_i)$ where $\{I_j \ | \ j \in J\}$ is a partition of I.

My problem is following: prove that $\Pi_{i \in I} \mathbf{A}_i \cong \Pi_{j \in J} (\Pi_{i \in I_j} \mathbf{A}_i)$, where $\langle \mathbf{A}_i \ | \ i \in I \rangle$ is an indexed set of similar ...
3
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33 views

Let $h: A \to B$ be a weak homomorphism. Is h$[A]$ a substructure of $B$?

A little bit more precise: let $\mathfrak{A}$ and $\mathfrak{B}$ be two structures. Define a weak homomorphism as a function $h: \mathfrak{A} \to \mathfrak{B}$ such that the folowing conditions are ...
3
votes
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93 views

Countable transitive $T \vDash ZFC-P$ with $\approx$ not absolute: why do we need $H(\aleph_3)$?

I am trying to solve Exercise IV.3.31 from Kunen's Foundations of Mathematics. I think I have a solution but I am confused by one of the hints. By request, here is the text of the exercise. ...
3
votes
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79 views

Examples of Jónsson Models

Let $T$ be a complete first order theory. Suppose that $M\models T$. Then, $M$ is said to be a Jónsson Model of $T$ if for all $N$, such that $N\prec M$ and $N\models T$, we have $|N|<|M|$ (Note ...
3
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74 views

Finding a finite model

Hello I am having difficulty with this question, I am not even sure what strategy one would go about proving something like this: Suppose $L$ is a language which includes an infinite list ...
3
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148 views

Defining “structured sets”

In his Notes on Set Theory (p. 44) Moschovakis defines: A structured set is a pair $U = (A,S)$ where $A$ is a set, the space of $U$, and $S$ is an arbitrary object, the frame of $U$. But even ...
3
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67 views

Question on a Theorem from Chang-Keisler's Model Theory concerning $\Sigma^0_n$ sentences

The Theorem is 3.1.11 and states that for $n>0$ the following are equivalent : $\phi$ is equivalent both to a $\Sigma^0_{n+1}$ and a $\Pi^0_{n+1}$ sentence. $\phi$ is equivalent to a Boolean ...
3
votes
0answers
143 views

Crankery: Is there a perfect inner model of ZFC?

In the book "Aspects of Vagueness", the article "The alternative set theory and its approach to Cantor's set theory" by A. Sochor proposes the following definition: We will say that a set universe ...
3
votes
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92 views

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 ...
3
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103 views

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 ...
3
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0answers
105 views

The Logic of Satisfiability?

I am aware of some study into the logic of provability. It is generally taken to be intermediate in strength between S4 and S5 modal logics. Is there corresponding study into something like the logic ...
3
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169 views

A question about a passage in Just/Weese's Basic Set Theory

I have a question regarding the following passage in Just/Weese (p 192), $\mathbf{V}$ denoting the cumulative hierarchy: "But consider the following situation: Where in "($\beta$)" does the ...
3
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83 views

An application of Descriptive set theory in Model theory.

In page 162 of D.Marker Model theory book he proved that the set $S_n(F,T)$ of all $F$-types realized by some $n$-tuple in some countable model of $T$ is analytic (this is with any $F$:= $countable$ ...
2
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32 views

When do finite linear orders have the same theory in MSO?

Let $\mathfrak A$ and $\mathfrak B$ be finite linears orders and $m<\omega$. Then we have $$\mathfrak A \equiv^m_{FO}\mathfrak B \quad\text{iff}\quad |A|=|B| \,\, \text{or}\,\, |A|,|B|\ge 2^m-1.$$ ...
2
votes
0answers
62 views

$\overline{\mathbb Q}$ and $\mathbb C$ same first order theory

How do you show that $\overline{\mathbb Q}$ (the algebraic closure of $\mathbb Q$) and $\mathbb C$ have the same first order theory over the signature $(0,1,+,\cdot)$?
2
votes
0answers
46 views

A universal formula is not equivalent to an existential formula

Suppose a formula is looks like the following: $\forall x_1 ... \forall x_n \alpha$ Where $\alpha$ is a formula free of quantifiers. And if $P$ is a 1-ary relation letter, then the formula ...
2
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0answers
29 views

Types linearly-ordered by deduction?

I'm wondering whether anyone has come across the following concept before: Consider a first-order language $L$ and a type $p$ over a theory $T$. I say that $p$ has a well-ordered filter-base if there ...
2
votes
0answers
69 views

Non-forking frames in AEC

Here http://shelah.logic.at/files/875.pdf on page 15, item 4 in the proof of 2.2.6, I would like to know why $S(M)\leq \lambda \times \lambda^+$. I understand that models in $K$ have cardinality ...
2
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0answers
71 views

Local embedding implies embedding into an ultraproduct

I am reading Gorbunov's "Algebraic theory of quasivarieties" and can't prove some statements, which are supposed to be obvious I think. At first, here are some definitions and notations. For a given ...
2
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53 views

On the back and forth conditions for a set of partial isomorphisms

I've recently begun reading Poizat's A Course in Model Theory and already in the first pages I had some doubts. One odd (not necessarily bad) thing is that he defines notions such as isomorphism only ...
2
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112 views

How large or small can the gap in Shelah's main gap theorem be, up to consistency?

Shelah's main gap theorem in model theory says: For each first order complete theory $T$ in a countable language if $I(T,\kappa)$ denotes the number of its models of size $\kappa$ then one of ...
2
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53 views

The composition of functors and algebraic structures

An algebraic structure can be viewed as a finite-product-preserving functor $X : \mathbf{L} \rightarrow \mathbf{C},$ where $\mathbf{L}$ is a Lawvere theory and $\mathbf{C}$ is a category. Thus given ...
2
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102 views

Semantic Proof of Tarski's Undefinability of Arithmetic Truth

A few years ago I took a logic course and I've since lost my notes. I seem to remember a very semantic proof of Tarski's theorem on the undefinability of arithmetic truth, one that didn't use the ...
2
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0answers
41 views

Books/papers on model theory in non-monotonic logic

I am working on a project whose object language is in non-monotonic logic. Since the project involves reasoning about the models, I am thinking of translating a non-monotonic problem into a ...
2
votes
0answers
105 views

Isomorphism of finite models

Let $\mathfrak A$ and $\mathfrak B$ are models of finite signature $\sigma$. Prove that $\mathfrak A$ and $\mathfrak B$ are isomorphic, if $\mathfrak A \equiv \mathfrak B$ and $\mathfrak A$ is ...
2
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77 views

Lowering the power of infinite model

I need to prove that for every infinite model $\mathfrak A$ of signature $\sigma$ exists model $\mathfrak B$ with attributes: $\mathfrak A \equiv \mathfrak B$. $\parallel \mathfrak B \parallel = ...
2
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136 views

Recursive non-standard models?

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