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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Real Closed Fields with Predicate for a Dense Subfield

Consider $M = (\mathbb{R};+,<, \times, 0, 1, K)$ where $K$ is a unary predicate which holds on $\mathbb{Q}$ (or any dense subfield of $\mathbb{R}$). Question: Is it true that the parametrically ...
3
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1answer
28 views

Proving the Downward Löwenheim-Skolem using monotonic operators

This is another exercise from Kees Doets Basic Model Theory. Here's the idea. It's well known that the downward Löwenheim-Skolem theorem follows as an easy corollary of the following lemma using ...
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2answers
63 views

Proving that $\langle \omega+1, \leq \rangle$ and $\langle \omega + \omega^*, \leq \rangle$ are not elementarily equivalent

This is an exercise from Kees Doet's Basic Model Theory. The basic idea is to show that an ordering of type $\omega + 1$ is not isomorphic to an ordering of type of $\omega + \omega^*$, where ...
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37 views

number of types if isolated types are dense

Let $T$ be a countable, complete theory, $M\models T$ and $A\subset M$. Now, Theorem 5.12.16 in Srivastava's book "A Course on Mathematical Logic, 2nd Edition" says that $S_{n}^M(A)$ must be countable ...
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68 views

Completeness for Infinitary Logic?

I have heard a rumor that there is a proof system for certain infinitary logics, given by Carol Karp (?) in her thesis, but I can't find a copy. The result that I'm told exists is the following: A ...
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2answers
94 views

Is there an embedding of $\langle \mathbb{N} \setminus \{0\}, \leq, \times, 1 \rangle$ into $\langle \mathbb{N}, \leq, +, 0 \rangle$?

This appears to be a common beginning exercise in model theory (I found it both in Chang & Keisler and also in Manzano's Model Theory). It's not difficult to see that there is an embedding of ...
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35 views

Model theory exam question:Show that if $M$ is isomorphic to $N$, then there is an extention to signature $S*$ so that $M*$ isomorphic to $N*$

So I had an exam today, and this question was really difficult for me: Let $S$ be a signature with $S=\{<,f\}$, $n_f=2$. $M=(\mathbb R;<,+)$ and $N=(\mathbb R_{>0};<,\cdot)$, $+, \cdot$ ...
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56 views

Quantifier elimination over rationals.

My question is concerned with a statement in Marker's Model Theory. The statement is that for formula $\phi(a,b,c)=\exists x(ax^2+bx+c=0)$, we cannot have a quantifier free formula $\phi'$ such that ...
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30 views

Elementary substructure and elementary equivalence

So the subject I'm stumbling around a bit is elementary equivalence and elementary substructures. So maybe you could help me sort things out. So the definitions as I know them. Structures $M, N$ ...
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1answer
25 views

Over signature $S= \{f \}$, $n_f=1$, find a formula $\phi$ so that for every structure, $M \vDash \phi \implies|M|$ is infinite

Over signature $S= \{f \}$, $n_f=1$, find a formula $\phi$ so that for every structure, $M \vDash \phi \implies |M|$ is infinite I'm trying to solve this for a really long time. I tried to perhaps ...
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2answers
66 views

Proving that for structure $M=(\mathbb R; +,<,0)$, the function $f(x,y)=x \cdot y$ is not a definable set in $M$

Proving that for structure $M=(\mathbb R; +,<,0)$, the function $f(x,y)=x \cdot y$ is not a definable set in $M$. Note: A function is called a definable set, if its graph, meaning the set ...
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2answers
30 views

Let $T$ be a theory over $S$, prove that $T \vdash \{\phi \rightarrow \psi \} \iff (T \cup \phi) \vdash \psi$

Let $T$ be a theory over $S$, prove that $T \vdash \{\phi \rightarrow \psi \} \iff (T \cup \phi) \vdash \psi$. So this seems very obvious intuitively. I'm just not sure what is the 'technique' to ...
3
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0answers
89 views

Application of the reflection theorem in ZFC

In a proof that a certain theory $T$ in conservative over $\textsf{ZFC}$, the author makes the following step: (Here $\Delta$ is a finite set of formulas such that $\textsf{ZFC}\vdash\Delta$.) ...
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0answers
41 views

For signature $s= \{ f\}, n_f=1$, show that if for structures $M, N$, $f^N, f^M$ have only one orbit, of infinite order, then $M,N$ are isomorphic

For signature $s= \{ f\}, n_f=1$, show that if for structures $M, N$, $f^N, f^M$ have only one orbit, of infinite order, then $M,N$ are isomorphic. So we know that for some $m \in M$, the set ...
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1answer
24 views

A substructure that is elementary equivalent to its structure, isn't an elementary substructure example mistake?

So we were given this example to show that substructure that is elementary equivalent to its structure, isn't necessarily an elementary substructure. Just so I'll be clear on the definition: A ...
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0answers
34 views

Questions concerning a specific given theory.

Let $L$ be the a language containing countably many unary predicates $\{P_i : i\in \mathbb{N} \}$ and countable many constants $\{a_{i,j} : (i,j)\in \mathbb{N}^2 \}$. Let $T$ be the theory which says ...
3
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1answer
54 views

Ultraproducts explicit example

so I'm studying for my exam in the Model Theory course I take. I just can't really wrap my head around ultraproducts. It's so abstract and complex. Is there a place where I can be shown with an ...
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1answer
36 views

$\kappa$-categoricity in the language of identity

I have the following exercise from Dirk Van Dalen's Logic and Structure: Here with language of identity we mean the language with no extralogical symbols (i.e. no symbols of predicates, functions ...
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1answer
72 views

Number of non-isomorphic models

Let $C$ be the class of cardinals. Define by recursion $C_0 = C$, $C_\alpha = C_\beta\cup P(C_\beta)$ if $\alpha=\beta+1$ and $C_\alpha = \bigcup_{\beta<\alpha}{C_\beta}$ for limit $\alpha$ (Here ...
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4answers
85 views

Non-standard model of $Th(\mathbb{R})$ with the same cardinality of $\mathbb{R}$

Let $\mathfrak{R}= ⟨\mathbb{R},<,+,-,\cdot,0,1⟩$ be the standard model of $Th(\mathbb{R})$ in the language of ordered fields. I need to show that there exists a (non standard) model of ...
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Why do we know that Gödel sentences are true in the standard model of set theory, but do not know if the continuum hypothesis is?

By what methods can we identify sentences that are true in the standard model of set theory, but not in other models? In particular, how do we prove that Gödel sentences are true in the standard ...
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60 views

General notions of “basis” in algebra/model theory

Free groups, free abelian groups, and vector spaces all have a notion of 'basis': a subset $B$ of the structure such that everything in the structure can be written uniquely as a finite combination of ...
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1answer
31 views

Hardness of index sets for computable structures

Suppose we have a computable structure $M$ and we want to show that its index set $I(M)$ is (many-one) $\Gamma$-hard for some complexity class $\Gamma$ (like $\Sigma^0_2$). To do this, we need to show ...
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Question concerning types, partial elementary maps and universality.

Show that the following are equivalent: $M$ realizes all $\lambda$-types of $M$ over $\emptyset$ for any $\lambda<\kappa$. For any $N\equiv M$ and any $A \subseteq N$ with $|A|<\kappa$ there ...
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1answer
32 views

On some equivalences of $\omega$-categoricity.

Let $T$ be a complete theory without finite models. Prove that the following are equivalent: $T$ is $\omega$-categorical. $T$ has a countable model which is both atomic and saturated. All models of ...
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0answers
27 views

Whether certain reducts of relational structures are themselves elementary classes?

Suppose I have a theory consisting of binary relation symbols $R$ and $S$. The theory is divided into a set of formulas involving only $R$ and equality, and also a single formula of the form ...
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1answer
38 views

Are A. Malcev's conditions first order?

There is a Russian paper by A. Malcev written in 1939 that give infinitely many jointly necessary and sufficient conditions for a (not necessarily commutative) monoid to be embeddable. Are those ...
2
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1answer
45 views

Simple examples of non-isomorphic but elementarily equivalent structures.

An uncountable example is obtained by considering dense linear orders without endpoints. Any two dense linear orders without endpoints are elemenatarily equivalent. But $\langle\mathbb{R},<\rangle$ ...
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Is the class of group-embeddable monoids an axiomatizable class?

Is the class of monoids which can be embedded in a group a first-order axiomatizable class? And if it is, is it finitely axiomatizable?
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1answer
33 views

Calculate rank of Morley in $ACF_{0}$

How I can calculate the Morley rank of the type $x = x$ in the theory $ACF_{0}$?
2
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1answer
66 views

Inclusions of Vector Spaces vs Sets

I have a conjecture relating statements about inclusions of sets to corresponding statements about inclusions of linear subspaces. More specifically, consider a formula \begin{equation*} \phi \equiv ...
1
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1answer
24 views

Sequence of indiscernibles in in saturated model.

Let $\lambda > \aleph_{0}$ and M a $\lambda$-saturated structure. Show that for every non-algebraic $\phi(x,a)$ with parameters from $M$ there exists a sequence of indiscernibles $I \subset ...
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2answers
41 views

Indiscernible to create descending chain of elementary models

Let $M$ an infinite structure such that $\mid M \mid \ge \mid L(M) \mid $. Show that exists a proper elementary extension $N$ and a chain $\langle N_{i} \mid i < \omega \rangle $ such that ...
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0answers
58 views

References for Introductory Model Theory focusing on applications other than algebra

I would appreciate suggestions for references (books, lecture notes, articles etc...) on Model Theory (at an introductory level) that don't focus mostly on algebra when giving examples and/or applying ...
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1answer
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Sequences of indiscernibles over sets.

Let $\mathbb{I}=\langle a_{i} \mid i< \omega + \omega\rangle$ be a sequence of indiscernibles over a set $A$. Then $\langle a_{i} \mid \omega \le i< \omega + \omega\rangle$ is an ...
4
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1answer
58 views

Is subset relation axiomatizable?

We know that the ZFC axioms define the elementary class associated with them. And we can extend the signature to a binary relation symbol P and add a defining axiom that says P is the subset relation. ...
3
votes
1answer
36 views

$a \notin acl(A) \Longleftrightarrow$ exists an infinite indiscernible sequence over $A$ containing $a$

Show that $a \notin acl_{\mathbb{M}}(A) \Longleftrightarrow$ exists an infinite indiscernible sequence over $A$ containing $a$. $ \Rightarrow) a \notin acl_{\mathbb{M}}(A)$ then $\forall ...
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1answer
44 views

Example for consistent set of sentences that is $k$-categorical and has infinite models

I'm looking for an example for a consistent set of sentences $T$ (in first-order logic) that is $\kappa$-categorical (so each two models of $T$ with cardinality $\kappa$ are isomorphic) and that has ...
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1answer
36 views

Proving the Overspill Principle.

The Overspill Principle, as I have encountered it, states: Given $M$ a non-standard (i.e. not isomorphic to the naturals) model of Peano Arithmetic, $\varphi$ a formula with $n+1$ free variables ...
2
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1answer
55 views

Vaught's two cardinal theorem using Vaught pairs

I've been reading David Marker's Introduction to Model Theory, and found Vaught's two cardinal theorem (4.3.34): if a theory $T$ has a $(\kappa,\lambda)$-model, where $\kappa > \lambda \geq ...
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1answer
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Hodges, motet Non vos relinquam and indiscernible sequences.

This image is taken from the book "A Shorter Model Theory" of Wilfred Hodges, page 250 in the beginning of Chapter 9, structure and categoricity. It is the beginning of the motet Non vos relinquam ...
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1answer
38 views

Definable in a theory $T$ strongly minimal

Let T strongly minimal and $\phi(x,y)$ a $A$-formula and $b$ an arbitrary element. Let $\mathbb{M}$ the monster model. Suppose $\phi(\mathbb{M},b)$ is quasi-definable over $A$. Show that ...
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1answer
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Definable and quasi-definable over models.

Definition: A set X is quasi-definable over C if is definable over all model M containing C. i) Show that a singleton $\{b\}$ is quasi-definable over C iff $b$ is algebraic over C ii) ...
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1answer
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Orbit of an element $a$ over a set $A$ in the monster model.

Let $\mathbb{M}$ the monster model of a theory $T$. If the orbit of an element $a$ over a set $A$ of elements reals, $O(a/A)$ is not finite then it must be of the size of $\mathbb{M}.$ In this ...
3
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1answer
30 views

Non-forking global types

Work inside a large saturated model ${\cal U}$. Let $p(x)$ be a global type that does not fork (=divide) over $A$. Let $\varphi(x,b)$ be a formula in $p(x)$. And let $\langle b_i:i<\omega\rangle$ ...
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1answer
74 views

Metrizability, Models, of Non-Standard Reals

according to compactness theorem in logic, there are models for the Reals of all infinite cardinalities, and these are elementary-equivalent to those of the "Standard" Reals ( Reals with ...
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Why is a type being realised in a finite structure enough to ensure it is isolated?

I'm looking for a proof that a type realised in a finite structure, modelling some theory $T$, is isolated (aka principal). Definitions for clarity are: Definition An $n$-type $p$ is realised in a ...
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1answer
117 views

The Hyperreal number system

Currently reading Infinitesimal Calculus by Henle and Kleinberg. In this text, page 25, they note that they define a hyperreal number system, not the hyperreal number system. This is because "there ...
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Elementary submodels in stationary logic.

In the paper "Stationary Logic" by Barwise, Kaufmann and Makkai the authors prove that stationary Logic L(aa) has Löwenheim number $\aleph_1$, i.e. every satisfiable set of sentences has a model of ...
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1answer
66 views

Prove or disprove wether the sentence $\exists x\forall y Q(x,y)\to \forall y\exists x Q(x,y)$ is logically true

I got stuck at this problem for some hours: Determine whether the first-order sentence $\exists x\forall y Q(x,y)\to \forall y\exists x Q(x,y)$ is logically true, where $Q$ is a 2-ary predicate ...