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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Is it possible to have logic without syntax (with only semantic proof methods)?

In one paper I have read a note "Thus, unlike approaches which make use of full first order logic, unprovability of a formulae with respect to a agent specification can be shown by each of two ...
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28 views

Embeddable rings axiomatic class?

In this question, a ring is defined to be with a unit distinct from the zero element, not necessarily a commutative ring though. Is the class of all such rings that can be embedded into fields an ...
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25 views

First-order logic: largest size among smallest finite models for formulas of a given length

Apologies for the somewhat cryptic title. For any first-order formula X, let ssm(X) be the size of the smallest finite model of X. By size I mean number of individuals. So, for example, ssm('Fa ...
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Is there a first order formula $\varphi[x]$ in $(\mathbb Q, +, \cdot, 0)$ such that $x≥0$ iff $\varphi[x]$?

In the first-order language $\mathscr L$ having $(+, \cdot, 0)$ as signature, it is easy to define a formula $\phi[x]$, namely $\exists y \; x = y^2$, satisfying : $$\text{for all } x \in \Bbb R, ...
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42 views

algebraic closure is the intersection of all elementary sub-models of the monster

This is a question from an exercise in model theory. Let T be a complete theory, $ \mathfrak{C} $ monster model of T (a $ \kappa $ saturated model of cardinality $ \kappa $ for some large $ \kappa $) ...
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35 views

Proof that the inverse limit of an inverse system is equal to another set

I'm trying to to learn model theory and so working with some basic examples. Consider the following: Let $D$ be finite subsets of $\mathbb{Q}$ with the ordering given by the subset relation. Let ...
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48 views

Theories of Arbitrary Morley Rank

Suppose that you have a language $L$. I can show that theories like DLO, or any unstable theory for that matter, has Morley Rank $\infty$. I can also show that $REI_\alpha$ has Morley rank $\infty$, ...
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44 views

$\omega$-categoricity and infinite languages

The Ryll-Nardzewski Theorem states that an equivalent condition to $\omega$-categoricity is that there is a finite number of $n$-types for any $n$. So what happens when you add a countably many unary ...
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61 views

Why do ultraproduct structures use a quotient as their universe?

For an $L$-structure $\mathfrak{A}$ with universe $A$, if we have an index set $I$, with an ultrafilter $U$, we create an ultraproduct structure having as its universe $\Pi_I \;A_i/U$. This is the set ...
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36 views

Are ($Q$, $\leq$) and ($Q \times Q$, $\leq _e$) isomorphic? [duplicate]

I can't really tell if ($Q$, $\leq$)$\cong$($Q \times Q$, $\leq _e$), where $\leq_e$ denotes the left lexicographic order. Neither have a last/first element, both are dense and have the same ...
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32 views

Extending the language in Henkin style completeness proof for first-order logic

There is a detail in the Henkin style proof of completeness for first order logic that I can't quite understand. So in the first part (Lindenbaum's Lemma), we need to show that a consistent set of ...
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664 views

Does a finite first-order theory which has a model always have a finite model?

I'm curious whether this is true or not: Let T be a finite, first-order theory. If T has a model, then T has a finite model. I would assume the answer is 'yes', but I wanted to make sure I ...
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51 views

Definibility of $\mathbb{Z}$ in product rings

If $R$ is a product ring whose factors are in a finite number and are all quotients of $\mathbb{Z}$ (that is, either $\mathbb{Z}$ or $\mathbb{Z}_n$'s ), is it a sufficient and necessary condition for ...
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1answer
63 views

Boolean model containing both confusion and junk

I'm doing a course in Equational Programming, and really new to these materials. So we got a specification for Booleans: ...
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1answer
83 views

Is $\mathbb Z$ first-order definable in (the ring) $\mathbb{Z\times Z}$?

Is $\mathbb Z$ first-order definable in $\mathbb{Z\times Z}$ (using sum and product but obviously not the concept of "component")? I believe no but how may I prove it? Is this standard?
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65 views

Is it possible to formalize $T \models \sigma$ within the deductive calculus?

I know that the notion of $T \vdash \sigma$ is formalizable within any sufficiently powerful theory $T$ but is $T \models \sigma$ formalizable as well? How is this possible if there are infinitely ...
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1answer
22 views

On the existence of finite substructures when sufficient chain conditions are met

Let $L$ be a language and $T$ and $L$ theory. Suppose that for any $M\models{T}$, we have $M\subseteq{\bigcup{C_{n}}}$, where each $C_{n}\models{T_{\forall}}$ is finite. I want to show that for some ...
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19 views

1-model complete

For $L$ structures $A$ and $B$ we write $A\preceq_{1}B$ if $A\subseteq{B}$ and $A\models{\varphi(a)}$ iff $B\models{\varphi(a)}$ for any finite tuple (of the correct length) $a$ from $A$ and for any ...
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81 views

Class models of $\mathsf{ZFC}$ and consistency results

First of all, I'm only starting to study independence results in set theory. And there is one obstacle that confuses me a lot. Probably such questions have already been asked, but I haven't found ...
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1answer
46 views

Indiscernibles over a model

Working within the framework of a monster model, I wish to show that: (*) If $(a_{i}:i<{\lambda})$ is an indiscernible sequence over $A$, then there is a model $M$ containing $A$ such that ...
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29 views

Definition of Local Finiteness

Let $L$ be a language and let $T$ be an $L$-Theory. $M\models{T}$ is said to be locally finite if for any given finite subset $X$ of $M$, there is a finite substructure $A$ of $M$ s.t. ...
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54 views

D.Marker's axiomatization of rings

Adding "-" as a binary function to the language of rings and the sentence $∀x(x+(−x)=0)∀x(x+(−x)=0)$ to the set of axioms proves existence of additive inverses. But I can't see how Professor Marker's ...
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64 views

Upward Löwenheim Skolem

I'm trying to understand the proof of (a version) of the upward Löwenheim Skolem Theorem, which states that given a language $\mathscr{L}$ and a set of $\mathscr{L}$-sentences $\Sigma$ with a ...
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64 views

Absoluteness and Extensionality

In the set theory text that I am reading, the author writes: Relative to the set $A = \{ 0, \{\{0\}\} \}$, the sets $0$ and $\{\{0\}\}$ are indistinguishable in the sense that $[$for all $x$ in ...
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53 views

elementary equivalence and incompleteness

I read the following line in a text on set theory: "Peano Arithmetic has continuum many non-isomorphic countable models (including the standard model omega), all of them elementary equivalent." ...
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1answer
61 views

ACF universal is the theory of integral domains

When studying David Marker's "Model Theory: An Introduction" book trying to understand the proof of Lemma 3.2.1 which says: $ACF_{\forall}$ is the theory of integral domains, I couldn't understand the ...
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1answer
48 views

A test for quantifier eliumination

In David Marker's "Model Theory: An Introduction" book I was trying to prove Corollary 3.1.12 which is left for the reader, but I couldn't reach any solution. The aforementioned corollary states: ...
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2answers
54 views

An AE axiomatization of groups

Let $L=\{*\}$. The usual axiomatization of groups in this language has the EA axiom $\exists{e}\forall{x}$ $ e*x = x$. But the union of a chain of groups is also a group. This means that the theory of ...
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0answers
56 views

Shelah's materialize vs. realize;note: tags are badly chosen due to the lack of them

Can someone please explain to me in some detail the exact difference between materialize and realize for a Galois type $p$? Esp. is realize a special case for materialize? Why is it so? What is the ...
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2answers
45 views

Clarify definitions of relation and 0-ary relation

From mathworld.wolfram.com: A relation is any subset of a Cartesian product But if so, then the null set is all of: 0-ary, 1-ary, 2-ary etc. Wouldn't it be better to define it as: A relation ...
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87 views

Definition for non-dividing

The definition for non-dividing is taken as the negation of the definition for dividing (as found in http://www.math.cmu.edu/~rami/simple.pdf : Definition 1.1 for example). Thus assuming ...
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55 views

When can independence of a statement in a theory be reduced to “truth”?

Since the Goldbach conjecture is in $\Pi_1^0$, if it were proven to be independent of Peano Arithmetic, it would follow that the Goldbach conjecture is true (i.e. true in the standard model), since ...
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39 views

Prove that the Morley Rank is preserved under definable bijections.

I need to prove this: If there is a definable bijection between $\varphi(C)$ and $\psi(C)$ then $RM(\varphi)= RM(\psi)$. Where $C$ is the monster model. I can intuitively understand it, the Morley ...
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43 views

prove that any two isomorphic structures are elementarily equivalent

Imagine we have two L-structures $M$ and $N$. For each L-sentence $\phi$ , $M$ models $\phi$ iff $N$ models $\phi$. We call $M$ and $N$ two elementary equivalent L-structures. We say $M$ and $N$ ...
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1answer
64 views

Understanding types and the proof that every type is realized in an elementary extension.

So I've been recently been studying types from David Marker's book and have some issues understanding them and in particular why did Marker choose to present certain proof of the following theorem ...
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1answer
68 views

Type of Infinite Tuple

Things along the following lines is often said about infinite tuples in model theory (we are assuming that we are working inside some monster model $M$ of some complete $L$ theory $T$): If I is a ...
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1answer
24 views

Is there a conservative extension of IZF that extends IZF by a weak form of the axiom of choice?

The full axiom of choice implies the LEM, and so is incompatible with constructive mathematics, although there are some weaker forms of the axiom of choice, such as the axiom of dependent choice, or ...
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42 views

Forking in Strongly Minimal Theories

I have been trying to define $A\overset{\vert}{\smile}_{C}B$ in a strongly minimal theory (let's say countable to avoid complications though I'm not sure if this matters). My attempt is based on the ...
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1answer
26 views

Intervals in divisible ordered groups

Is it true that if $(G,+,0,<)$ is a divisible ordered abelian group with at least two elements, then for $a,b >0 \in G$, there is an injective order preserving map from $[0;a)$ to $[0;b)$? It ...
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1answer
46 views

Representing a $\sigma$ - structure using a signature-$\sigma$ in Mathematical Logic.

In mathematical logic, I have a question regarding how a signature-$\sigma$ relates to a corresponding $\sigma$ structure which interprets the signature-$\sigma$ In Chiswell and Hodges book ...
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Explicit countable elementary extension of $\mathbb{N}$

I would like to see an explicit example of a non-trivial elementary extension of the structure $(\mathbb{N}, +, \cdot, 0, 1)$ where $\mathbb{N}$ includes zero. Most of all I am interested in countable ...
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75 views

How does Gödel's second incompleteness apply to any theory containing arithmetic?

If I understand correctly, there are two facts proven by Gödel's second incompleteness theorem, for a formal theory containing arithmetic 1) It is possible to express the consistency of the theory ...
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2answers
35 views

To build a tree in a countable theory with uncountable many types in finite variables

I'm trying to prove that if $T$ is a countable consistent theory that has no binary trees, then $T$ is small. i.e $|S_n (T)|=\aleph _0$ In order to do this i assume toward contradiction that $S_n ...
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If $\mathcal{T}_1$ and $\mathcal{T}_2$ admit quantifier elimination, does $\mathcal{T}$ admit quantifier elimination?

Let $\mathcal{T}_1$ and $\mathcal{T}_2$ be theories with disjoint signatures $\mathcal{L}_1, \mathcal{L}_2$. Form a new language $\mathcal{L} = \mathcal{L}_1 \cup \mathcal{L}_2 \cup \{P_1, P_2\}$, ...
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127 views

A structural view to the power set axiom: Is this axiom really justifiable?

The power set axiom in set theory states that the collection of the subsets of a set is a set itself. I wonder if this is a "natural" axiom in the sense that if we consider sets as the simplest ...
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1answer
69 views

How do we know $\mathbb{N}$ is a model of Peano Arithmetic?

The induction axiom in the theory of Peano Arithmetic (PA) is actually an axiom scheme such that for every formula $\phi(x,\bar{y})$ with free variables $x,\bar{y}$ ($\bar{y}$ being a string of ...
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35 views

Hodges exercise 2.7.1: Quantifier elimination in dense linear orderings

In Hodges' A Shorter Model Theory, exercise 2.7.1 tells you to prove theorem 2.7.1, which says that the following five formulas are an elimination $\Phi$ set for the class of all dense linear ...
3
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1answer
34 views

two different realization of one type

Im having trouble solving this problem: Let $M$ be a saturated structure of cardinality $\kappa$. Let $A\subseteq |M|$ with $|A|<\kappa$. Then there is a type $p\in S_1(A)$ with two different ...
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80 views

What does the theory of the empty set look like?

I know next to nothing about logic, but I was wondering what first order axioms would give rise to the theory of the empty set (that is to a theory whose only model is the empty set)? The problem I ...
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138 views

Detecting incomparability in countable elementary submodel

This might be just an easy exercise in model theory but I can't seem to wrap my head around right now. Let $\theta$ be large enough regular cardinal and $\kappa < \theta$. $(\kappa, \prec)$ is ...