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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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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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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43 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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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. ...
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$a \notin acl(A) \Longleftrightarrow$ 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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74 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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55 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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72 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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119 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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42 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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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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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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35 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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94 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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164 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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85 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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63 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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83 views

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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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 ...
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65 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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61 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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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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69 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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54 views

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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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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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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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 ...
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60 views

show that a horn sentence is preserved under a direct product.

show that a horn sentence is preserved under a direct product. If $\varphi$ is a horn sentence and $\mathfrak{A}_i, i \in \text{I}$ is a model for $\varphi$ namely $\mathfrak{A}_i \vDash \varphi$ ...
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51 views

Indiscernibles and colorations.

Let $L$ be a language, and $(X,\leq)$ be a total order contained in an $L$-structure $\frak{A}$. Now if we denote by $[X]^n$ the set of $n$-sized sequences in $X$ and consider a set $\Gamma$ of ...
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Is algebra needed to really understand and/or enjoy model theory?

What are the desirable pre-requisites to be able to learn model theory well? In particular, it seems that connections to algebra are used heavily especially as examples. I would like to know if a ...
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47 views

Proving that every submodel of DLOE is an elementary submodel

Let $A$ and $B$ be models of the theory of Dense Linear Orders without Endpoints such that $|B| \subset |A|$. I'm trying to prove that $B$ is an elementary submodel of $A$. Using Tarski-Vaught test I ...
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51 views

Homogeneous models in a strongly minimal theory

I am trying to prove that every infinite model of a complete strongly minimal theory T is homogeneous. Clearly, if $k:M \to M $ is a partial elementary map with $|k|<|M|$ and $M$ is a model of ...
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How Many Countable Models of the successor function

Consider the successor function (s(x)=x+1), $T_{S}$ to be the set of axioms given by; S1: ∀xy[s(x)=s(y)→x=y] (injective) S2: [s(x)≠0] (never 0) S3: ∀x[x≠0→∃y[s(y)=x]] (everything bar 0 is in image) ...
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On proving the zero-one-law for first order logic

I'm trying to understand the proof of the zero-one-law for first order logic as provided in (Ebbinghaus-Flum, 1995). It goes as follows: Let $\tau$ be a relational signature. Let $r\in\mathbb{N}$, ...
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Construction of an elementary extension satisfying $Th(\mathcal{M}) \cup S$ where $S(x)$ is an arbitrary set of formulas

Let $S(x)$ be a set of $\mathcal{L}$-formulas (containing at most the free variable $x$). Is there an elementary extension $\mathcal{N}$ of $\mathcal{M}$ such that $\mathcal{N} \models Th(\mathcal{M}) ...
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Deductive closure of sentence $\forall x \forall y F(x,y) \stackrel{.}{=} F(y,x)$ in language $\mathcal{L}$ is undecidable.

$\mathcal{L}$ is the language that contains a single binary function symbol $F$. In the earlier parts of this question, we were told to take the $\mathcal{L}$-structure $\mathcal{M}$ with universe ...
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114 views

Non-standard model of arithmetic and Gödel's theorem [closed]

This is a cross-post of a question asked on History of Science and Mathematics Stack Exchange. I've read Skolem's paper on his non-standard models of the arithmetic ("Über die ...
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Models and signatures for propositional logic

The following is a bit long, so I collected my questions at the end, but as this is the only opportunity I get for feedback I would appreciate it if anyone could also point out where I've gone astray ...
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What is the degree of a real closure of an ordered field?

Given a subfield $F$ of $\Bbb R$, let the real-closure of $F$ be the smallest subfield of $\Bbb R$ which is real-closed and extends $F$. Since the theory of real-closed fields is complete, this is in ...
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Models of the successor function

I would like to ask a few questions about models of the succesor function (s(x)=x+1), intact that is a bit vague, consider $T_{S}$ to be the set of axioms given by; S1: $\forall xy[s(x)=s(y) ...
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61 views

Axiomatizability in monadic second-order logic

For my thesis in finite model theory I'm considering some basic classes of structures, and I want to show in which logical systems they can or cannot be axiomatized. I now consider the class ...
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Every theory eliminates quantifiers in an appropriate definitional expansion?

I need to prove that every theory eliminates quantifiers in an appropriate definitional expansion. For this, consider: let $T$ be a theory in language $L$. Consider the following expansion of the ...
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Characterization of superstability

In a stable theory every global type $p$ is invariant (= non-forking) over ${\rm acl^{eq}}(A)$ for some set $A$. Is there a characterization of superstability and/or $\omega$-stability in terms of the ...