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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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 $DLO\vDash\neg\psi$ ...
3
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0answers
125 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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1answer
50 views

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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1answer
176 views

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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2answers
126 views

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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479 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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2answers
185 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
48 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 ...
7
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5answers
524 views

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
51 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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0answers
39 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
54 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 ...
4
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4answers
173 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 ...
4
votes
4answers
126 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
votes
1answer
86 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
votes
1answer
109 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 ...
3
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0answers
93 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 ...
6
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2answers
155 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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5answers
209 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
votes
2answers
187 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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2answers
285 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 ...
2
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1answer
84 views

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 ...
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1answer
191 views

Questions about the concept of Structure, Model and Formal Language

When we start to define mathematical logic (specifically, propositional, first order, and second order logic) we start defining the concept of a language. At the begining this is done in a purely ...
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0answers
116 views

the amalgamation property

Definitions: The age of a structure $M$ is the class of finitely generated substructures of $M$. A class of structures K has the amalgamation property (AP) if Whenever $A,B,C$ belong to $K$ ...
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2answers
349 views

Is the compactness theorem (from mathematical logic) equivalent to the Axiom of Choice?

Or more importantly, is it independent of the axiom of choice. The compactness theorem states the given a set of sentences $T$ in a first order Language $L, T$ has a model iff every finite subset of ...
3
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1answer
146 views

Model theory/stability theory

I was flipping through Baldwin's Stability Theory book and saw an example that has me confused... The example is a 1st-order theory $T$ of refining equivalence relations $E_i(x, y), i< ω$, where ...
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2answers
268 views

Why continuum hypothesis implies the unique hyperreal system, ${}^{\ast}{\Bbb R}$?

On page 33, Robert Goldblatt, Lectures on Hyperreals(1998): Now it has been shown under certain set-theoretic assumption called continuum hypothesis the choice of $\mathcal F$ is irrelevant: All ...
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1answer
51 views

Action of automorphisms on Nonforking Extensions

Let $T$ be a totally transcedental theory and $M$ a model of $T$ with $A \subset M$. Let $p(x) \in S(A)$ (a complete type with parameters in $A$) and let $q(x)$ be a nonforking extension of $p(x)$ ...
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2answers
413 views

What is “induction on complexity of formula”

The fundamental theorem of ultraproducts Given a language $L$ and a family of models $M_i$ of $L$ and ultrafilter $\mathcal U$ on $I$ and $\varphi$ a formula of $L$ then $$ \prod_{i \in ...
2
votes
1answer
63 views

Constants in ultraproduct are well-defined

When defining ultraproduct it is defined to be product of domains of models $A_\xi$ modulus the equivalence relation by ultrafilter on index set. The relation on the product are defined by ...
2
votes
1answer
55 views

Elementary theory of an algebraic structures

Could someone elaborate me what the sentence "The elementary theory of finite fields is decidable" means? I'm not sure that for example if I take $x\in \mathbb{F}_4$ and $y\in \mathbb{F}_5$ then can I ...
2
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2answers
64 views

Finite ultraproduct

I stucked when trying to prove: If $A_\xi$ are domains of models of first order language and $|A_\xi|\le n$ for $n \in \omega$ for all $\xi$ in index set $X$ and $\mathcal U$ is ultrafilter of $X$ ...
2
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1answer
63 views

About Ehrenfeucht's theorem proof

I am re-reading the proof of Ehrenfeucht's theorem on page 90 of Mathematical Logic of Shoenfield. I have the following problem. For the sake of clarity I have highlighted in red the problematic part ...
6
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3answers
155 views

There is concept of finite sets that can have only one “interpretation”?

In our mind we have a naive idea of what a set is, and in nature we can only observe something that behave like a finite set, ZFC (or set theories in general) tries to catch these properties in ...
7
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1answer
100 views

What's more robust than a structural homomorphisms?

This question isn't category theory; but, category theoreticians tend to be interested in mathematical structure, so I thought the answer might exist within that knowledge base. Given two ...
4
votes
1answer
112 views

What is the formulation of the Least Upper Bound propierty in First Order Logic?

I've been readining about the completeness Godel's theorems. Accordingly, the axioms of $R$ in first order logic make up one of these sets that is complete and consistent. But I've always seen the ...
3
votes
2answers
363 views

Is Euclidean Geometry complete and unique

Please help me understand this concept of completeness of geometry and set me on the right path. This is my context: From wikipedia, a formal system is complete if every tautology is also a theorem. ...
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244 views

Number of automorphisms of saturated models

I have the following assignment question: Let $M$ be an $L$-model of cardinality $\kappa$. Assume $M$ is saturated. How can you show that $|\text{Aut}(M)|=2^{|M|}$? I see two possible ...
3
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1answer
61 views

Trying to define an abstract notion of a function that turns sets of sentences into the set of models satisfying those sentences.

Let $\mathsf{Sen}$ denote a Boolean algebra, thought of as a collection of sentences, and let $\mathsf{Mod}$ denote a set without any additional structure, thought of as a collection of models. I'm ...
3
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1answer
119 views

Model theory question with finiteness

It should be pretty elementary, but I can't really see it.. Cheers to anyone who can help me Let $\Psi$ be a transitive set that is a model of $ZF$. Then, $a$ is finite if and only if $\Psi \models ...
5
votes
2answers
77 views

How to show $\chi_{{}^{*}P} ={}^{*}\chi_{P}$ by transfer principle?

Let $\mathfrak{R}$ be the real number system, $(\mathbb{R},+,\cdot,<)$ and ${}^{*}\mathfrak{R}$ be the hyperreal number system $({}^{*}\mathbb{R},{}^{*}+,{}^{*}\cdot,{}^{*}<)$. Transfer ...
26
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4answers
1k views

Is $\mathbb{N}$ impossible to pin down?

I don't know if this is appropriate for math.stackexchange, or whether philosophy.stackexchange would have been a better bet, but I'll post it here because the content is somewhat technical. In ZFC, ...
3
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1answer
179 views

Why every member of ${}^{*}\mathbb{R}$ is infinitely close to some member of ${}^{*}\mathbb{Q}$

Let $\mathbb{Q}$ be the set of rational numbers. Show that every member of ${}^{*}\mathbb{R}$ is infinitely close to some member of ${}^{*}\mathbb{Q}$. This is an exercise on page 180, A ...
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70 views

Is a “model” only a proper model if everything in it's definition is also explicitly constructed?

Say you have some collection of axioms and you find a set $X$ fulfilling them. And the definition of the set $X$ involves another concept, e.g. the real number, but only in a way which refers to the ...
3
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1answer
85 views

construction set of natural number logic

I identify the natural number $0$ with the empty set $\emptyset$, $1$ with $S(0)$, $2$ with $S(1)$, etc, etc. The axiom of infinity says $\exists x (\emptyset\in x\wedge \forall z\in x\space ...
5
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1answer
130 views

Inherited topology of logical Stone's spaces.

I'm asking here if the following construction is of any interest. I can not find any reference for that kind of thing, so either the subject is completely trivial, either I just don't have the correct ...
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1answer
211 views

Godel number and expressibility [duplicate]

how to show that these properties of strings of symbols are expressible: 1) being a term, 2) being a formula 3) being a sentence 4) being a proof in PA and where a property (i.e., predicate) P of ...
5
votes
1answer
113 views

Which logic is stronger? SOL or $\frak{L}_{\infty,\infty}$?

Both second-order logic($\mathsf{SOL}$) and infinitary first-order logic $\frak{L}_{\infty,\infty}$ are proper extensions of first-order logic($\mathsf{FOL}$), that is, are extensions of a ...
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2answers
129 views

What counts as a standard model of arithmetic?

In my research so far, I've found that the canonical standard model of arithmetic is $\mathbb{N}$ under the addition and multiplication operations. However, I've been unable to find much on any other ...
5
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0answers
108 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 ...