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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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 ...
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51 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
33 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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61 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
79 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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1answer
51 views

General notions of basis

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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29 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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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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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 ...
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1answer
39 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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1answer
20 views

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
32 views

Calculate rank of Morley in $ACF_{0}$

How I can calculate the Morley rank of the type $x = x$ in the theory $ACF_{0}$?
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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 ...
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1answer
23 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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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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32 views

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 ...
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1answer
51 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
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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
40 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
31 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 ...
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1answer
52 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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111 views

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
37 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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30 views

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
32 views

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 ...
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1answer
29 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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71 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
111 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
64 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 ...
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1answer
38 views

Prove that iff a formula $\phi (v_1, v_2,…v_n)$ is satisfied in the substructure $\mathcal M$, then it is satisfied in structure $\mathcal N$

Assume $\mathcal M \subseteq N$ structures for signature $S$. $\mathcal M$ is a substructure of $\mathcal N$. Let $\phi(v_1, \cdots v_n)$ be a formula without quantifiers. Prove by induction on ...
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1answer
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Hanf Numbers and Decidability

Currently reading J.L. Bell's Models and Ultraproducts and at the end of Chapter 4 section 4 the authors comment that "In spite of the fact that most languages can easily be shown to possess Hanf ...
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1answer
51 views

Generalizations of pregeometries

Combinatorial geometries and pregeometries are important in classifying strongly minimal (as well as O-minimal) theories. More formally, a model of a strongly minimal (or an O-minimal) theory with the ...
3
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1answer
59 views

Show that the set of all points $x \in \mathbb R$ where $f$ is differentiable is definable in $\mathcal M=(\mathbb R; +,-(), \cdot, \lt, 0,1,f)$

For the structure $\mathcal M=(\mathbb R; +,-(), \cdot, \lt, 0,1,f), n_f=1 $ show that the set of all points $x \in \mathbb R$ where $f$ is differentiable is a definable set. My issue here is how to ...
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2answers
58 views

Vacuously true? Prove or disprove that for every theory $T$, if $T$ is not satisfiable then for every $\phi$, $T \vdash \phi$

Is it vacuously true? Prove or disprove that for every theory $T$, if $T$ is not satisfiable then for every $\phi$, $T \vdash \phi$. If $T$ is not satisfiable, then there is no structure ...
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2answers
41 views

Prove or disprove that for theory $T$, $T \vdash (\phi \rightarrow \psi) \iff T \cup \{\phi\} \vdash \psi$.

Prove or disprove that for theory $T$, $T \vdash (\phi \rightarrow \psi) \iff T \cup \{\phi\} \vdash \psi$. This seems quite right, but I don't know how to prove it. So lets start with ...
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1answer
52 views

A singleton set $\{g\}$ can be regarded as a unary relation in $G$. Why?

Theorem 1.1. A relation $R \subseteq M^n$ is definable if and only if every automorphism of every elementary extension of $M$ preserves $R$. For a proof, the reader can see [4]. Suppose we ...
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Functions or relations stable under automorphism

Suppose we have a structure $M$, that is, a set $S$ with some designated functions and/or relations on that set. We can define automorphisms for this structure. What is the term in the standard logic ...
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Saturated models and $\kappa=\kappa^{<\kappa}$

Do not assume GCH. Can you characterize the cardinals $\kappa$ such that every theory $T$ with an infinite model has a saturated model of cardinality $\kappa$? I guess these are the cardinals such ...
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3answers
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Ordered field of rationals axiomatizable

Is there a set of sentences in the language of ordered fields whose models are precisely the rationals and any ordered field isomorphic to them?
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Reflexive reduct of preorder

Suppose P is a preorder on a set S, a reflexive and transitive relation. Suppose we subtract from P the identity relation and get a relation Q on S. Is the class of all such relations a first-order ...
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25 views

Definable valuation ring

If $(K,v)$ is henselian and $O_v$ is $\phi$-$\text{definable}$, why do I have that if $L\equiv K$ (in the language of ring) then $L$ admits a non-trivial henselian valuation ring? I understand if ...
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3answers
101 views

Complete extensions of a consistent theory

I understand that I need to use compactness but somehow can't finish it. Suppose $L$ is a language and $T$ a consistent $L$-theory with only finitely many logically inequivalent complete extensions. ...
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Is every reduced $k$-algebra all of whose residue fields are $k$ finitely generated?

Let $k$ be a field (of characteristic zero if you want). Let $A$ be a reduced $k$-algebra with the property that for every prime ideal $\mathfrak{p}$ of $A$ the natural homomorphism $k \to A/ ...
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Relative Identity vs Set Theory

The last complete, but unpublished, paper by the late Tom Etter titled "Three-Place Identity" purports to prove that all of mathematics can be expressed in terms of relative identity. In his own ...