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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Spectrum restrictions in the signature consisting of just a single binary operation

In the signature {*}, where * is an operator of arity 2, is there a theory whose spectrum is the set of prime powers?
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(stability-theoretic) ¨weakly normal groups" are closed under subgroups

Let me first introduce two definitions: For a structure $\mathcal{M}$ in a language $\mathscr{L}$ and a subset $X \subseteq M^n$, the fully induced structure on $X$ is a structure $\mathcal{X}$ with ...
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1answer
33 views

Why do we need ultrafilters to make sense of the cartesian product of $\mathcal{L}$-structures

I'm trying understand why we need ultrafilters in model theory. Here is how I see things. Could someone tell me if this is correct ? Further explanations are always welcome. Let $\mathcal{L}$ be a ...
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Multiplicative reducts of fields an elementary class?

Consider the multiplicative reducts of fields, that is fields except the addition operation is removed. We are considering the signature {*}, where * is an operator of arity 2. Is that class an ...
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Complete $n$-types for the theories of $( \mathbb Z , s )$ and $( \mathbb Z , s , < )$

This is exercise 4.5.2 from Marker's Model Theory: An Introduction (p.163), quoted verbatim: Let $T$ be the theory of $(\mathbb Z,s)$ where $s(x) = x+1$. Determine the types in $S_n(T)$ for each ...
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Saturated model for Th(Z,+,-,0,1)?

How an $\omega$-saturated model for the theory T=Th(Z,+,-,0,1) is made ? Can you give me some concrete example?
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1answer
114 views

Theories with countably many countable models

Having another question in mind (which is not yet fully worked out, but will come soon) I'd like to gather some examples of (interesting) theories with countably many countable models ...
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1answer
54 views

Show This theory is complete with four countable models

Let $\mathcal L_3 = \{<,c_o,c_1,\ldots\}$, where $c_o, c_1,\ldots$ are constant symbols. Let $T_3$ be the theory of DLO with sentences added asserting $c_o < c_1 < \ldots$. I would like to ...
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Show DLO + $\{ \phi : \phi = c_o < c_1 < \ldots \}$ has three models up to isomorphism

Let $\mathcal L_3 = \{<,c_o,c_1,\ldots\}$, where $c_o, c_1,\ldots$ are constant symbols. Let $T_3$ be the theory of DLO with sentences added asserting $c_o < c_1 < \ldots$. I would like to ...
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2answers
29 views

Morley Rank of Conjunction

Let $M$ be an $L$-structure. Let $\varphi ( x )$ and $\psi (x)$ be $L_{ M }$-formulas, where $x$ is some finite tuple of variables. With $\mbox{RM}$ we mean the Morley rank with respect to $M$ and ...
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1answer
59 views

Counterexample to Fraissé's Theorem for infinite signature

Let S be a finite signature and $\mathfrak{A}, \mathfrak{B}$ S-structures. Fraissé's Theorem states: $$\mathfrak{A} \equiv \mathfrak{B} \Leftrightarrow\mathfrak{A} \cong_f \mathfrak{B}$$ Where ...
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elementary substructure in a satureted model

Suppose you have a saturated model N of a complete theory T without finite models. How is it possibile to construct a proper saturated elementary substructure of N of the same cardinality of N ?
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36 views

Why do we tell functions from relations in structures?

A relation is a set of ordered pairs (a,b) A function is a relation (a,b) which satisfies the following conditions: For all a, there is one and only one b Therefore, all functions are relations. ...
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140 views

Compactness Theorem explanation

Compactness Theorem definition: If $T$ is a theory in a first-order language $L$, then $T$ has a model iff every finite subset $S$ of $T$ has a model. A number of questions regarding this ...
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1answer
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Prove a lemma on algebraic closure (model-theory)

Some help to prove this lemma? Let be N a saturated model. Let be $ \phi(x) $ a consistent formula with parameters in $A$ and $B\subset N$ a finite set . If for all $a$ such that $ \phi(a) $ there ...
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1answer
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algebraic closures (model -theory) [closed]

I try to prove this lemma : Let be N a saturated model. Let be $ \phi(x) $ a consistent formula with parameters in $A$ and $b\in N$ . If for all $a$ such that $ \phi(a) $ I have $b\in acl(A,a), ...
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1answer
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Model where all the infinite definable sets are of maximal size

Given a theory $T$ over a countable language with infinite models, and $\kappa$ an infinite cardinal, we can find a model of $T$ of size $\kappa$ whose infinite definable sets are all of size ...
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1answer
356 views

What is the real meaning of Hilbert's axiom of completeness

According to Greenberg's book of geometry it is sufficient to consider the axiom of Dedekind along with Hilbert's axioms (except of course for the Archimedian Principle and his Axiom of Completeness) ...
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Infinite palindromes in number of nonisomorphic posets is independent of $\mathsf{ZF}$

The following is an "exercise" in P. Stanley's book "Enumerative Combinatorics": Let $f(n)$ be the number of nonisomorphic $n$-element posets (...) let $\mathcal{P}$ denote the statement that ...
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1answer
24 views

Is the following structure $\omega$-categorical?

I am trying to figure out whether the following structure is $\omega$-categorical. The language contains countably many binary relations $E_n$ and a binary relation $<$. The structure itself is a ...
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Let $M,N$ be structures with relation $E$. $E^N$ and $E^M$ are equivalence relations, find sufficient and necessary condition for isomorphism

Let there be signature $S=\{E\}, n_E=2$, and let $M,N$ be S-structures. $E^N$ and $E^M$ are equivalence relations, find a sufficient and necessary condition for $M$ and $N$ being isomorphic. I ...
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1answer
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Model theory: Find an example for an infinite structure with only finite substructures

So I tried solving this for a long time: Find an example for an infinite structure with only finite substructures. So I tried looking at group signatures and infinite groups, but couldn't find an ...
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1answer
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A Characterization of Categories with a Conservative Forgetful Functor to SET

Examples of categories over $\bf{Set}$ such that the forgetful functor is conservative include the "algebraic" categories of groups, rings, modules, monoids, etc., but does not include the ...
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if $\mathfrak{B} \vDash BA$ then $S(B)$ is a stone space.

if $\mathfrak{B} \vDash BA$ then $S(B)$ is a stone space. proof: i show that $S(B)$ is a compact and hausdurf 1)$S(B)$ is a compact? i show that every cover of $S(B)$ has a finite subcover. i ...
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When you name an element in an uncountably categorical theory…

When you name an element in an uncountably categorical theory $T$ does it remains uncountably categorical? In other words, given a finite elementary map $f:M\to N$ between models of an uncountably ...
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3answers
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Why are $\vdash$ and $\vDash$ symbols from metalanguage?

I've read in some textbooks that $\vdash$ and $\vDash$ are symbols present only in metalanguage. From this, I infer that their use in object language is unacceptable. I would like to know why. Can't ...
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1answer
34 views

Random graphs are not uncountably categorical

Is there a simple proof that the theory of random graphs is not $\lambda$-categorical for uncountable $\lambda$?
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No proposition $\chi$ such that $\mathscr{M}\models\chi\iff\mathscr{M}$ is infinite

Let notation "$\models$" be used for the two following case: let $\mathscr{M}\models\varphi$, where $\mathscr{M}$ is an interpretation model and $\varphi$ is a proposition, mean that $\varphi$ holds ...
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1answer
31 views

Does elementary equivalence imply L-equivalence for structure L?

Does elementary equivalence imply L-equivalence for structure L? In the textbook "A Shorter Model Theory" by Wilfrid Hodges, page 39 defines both of these terms but does not tie them together. I was ...
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Does $T \models \forall \bar v (\phi(\bar v) \leftrightarrow \forall \bar w \psi(\bar v,\bar w))$ implies this formula?

Finishing the title, Does $$T \models \forall \bar v (\phi(\bar v) \leftrightarrow \forall \bar w \psi(\bar v,\bar w))$$ implies $$T \models \phi(\bar v) \leftrightarrow \exists \bar w \psi(\bar ...
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1answer
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How to prove an equality in a Lindenbaum-Tarski algebra?

Let $\mathscr{L}'= \mathscr{L}\cup \mathscr{C}$ be an extension of the language $\mathscr{L}$ with a new infinite set of constants $\mathscr{C}$, and $T$ be an $\mathscr{L}$ theory. I wish to show ...
2
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1answer
73 views

Very Simple Model Theory

I'm working through the fifth edition of Dirk van Dalen's 'Logic and Structure' and got stuck in section 4.3 on model theory. Let a structure (of some type) be a tuple $ \mathfrak{A} = (A; R_1, ...
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1answer
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Why isn't second-order ZFC categorical?

This builds off of (and reuses some of the examples from) this question about failures of categoricity not related to size. Second-order $\mathsf{ZFC}$ isn't categorical, but it's almost-categorical, ...
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Proof that $\cal K$ is an elementary class iff $\cal K$ is closed under elementary substructures.

Assume $\cal K$ is a pseudo elementary class. I need to prove that $\cal K$ is an elementary class iff $\cal K$ is closed under elementary substructures. Pseudo elementary class is a class of reducts ...
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1answer
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No countable models

I want an example of a theory T with finite models of arbitrarily large size but T has no countably infinite model. I know that T has to be uncountable, but couldn't come up with an example. ...
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embdedding standard models of PA into nonstandard models

Maybe it's well known to experts, but is there always an embedding $f$ of the standard model of Peano arithmetic into a nonstandard model? By Peano arithmetic I mean its first-order version, with the ...
4
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1answer
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$\kappa$-saturated, $1$-types - $n$-types

Definition. Let $\kappa$ be an infinite cardinal. We say that an $L$-structure $\mathfrak{A}$ is $\kappa$-saturated iff all $1$-types over sets of cardinality less than $\kappa$ are realised in ...
4
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2answers
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stratification (typage) of logic and syntax at the same time: is such a dream feasible? [closed]

This post is more philosophical than formal, yet I think it's an important question. There's an idea I have for long times already that would consist, in some sense, in doing a "theory of theories". ...
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1answer
55 views

Failures of categoricity not related to size?

The paradigmatic cases I know of theories failing to be categorical are cases that seem tied to the language's inability to distinguish between different infinite cardinalities of the theories domain. ...
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Show there is no elementary extension of $\mathbb{N}$ with an element between $0$ and 1

I have been presented with the follwing question and i want to see if the method i have used works, i have my doubts. We recall that M is an elementary extension of $N= \langle \mathbb{N}; +, ., 0, 1 ...
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prove Lindenbaum’s lemma for a countable language

Been reading through some model theory and got to a section on constructing models from syntax and i have been presented with the following problem, sorry for the lack of solution i just have no idea ...
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0answers
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What is the name of the set models can be drawn from?

What is the name of the set models can be drawn from? For example in propositional calculus an assignment function $v : P \rightarrow \{T,V\}$ can be the model of a formula $a$. What is the (generic) ...
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1answer
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Number of Ways of Combing Linear Orders

I have a slight variant of this question. I would also appreciate any references for questions like this. (The question is inspired by the study of linear orders in model theory.) Suppose you're ...
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1answer
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Walk me through this proof that a theory is satisfiable.

Setting Suppose $\mathcal M, \mathcal N \models T$, $\mathcal M \subseteq \mathcal N$, $\mathcal M$ existentially closed, then I I want to prove that there is $\mathcal M_1 \models T$ so that ...
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1answer
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Non Archimedean countable models of the theory of the reals

The questions in model theory I am trying to tackle is: Show that there is a countable model of $Th(\langle \mathbb{R};+,.,-,-,1,< \rangle) $ which is non archimedean. Honestly i dont really know ...
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complete, finitely axiomatizable, theory with 3 countable models

Does it exist a complete, finitely axiomatizable, first-order theory $T$ with exactly 3 countable non-isomorphic models? A few relevant comments: There is a classical example of a complete theory ...
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Models of the empty theory T, and proof that T $\kappa$ categorical for every cardinality. [duplicate]

Bombarding stack exchange with model questions today I am tackled with the following problem: Note this is the same question as posted by B0bg0blin's here, i just need a bit more clarity. In the ...
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2answers
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A fragment of Exercise 1.3.4 in _Shorter Model Theory_ by Hodges

The following is what I believe is necessary to solve Exercise 1.3.4 in Shorter Model Theory by Hodges. Given two structure $\mathcal {A, B}$ of the same signature $\tau$, a set $S$ of generators of ...
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First order theory of abelian groups and first order theory of cyclic groups are coincide?

Let $T$ be a first-order theory of cyclic groups. Even if an abelian group $(G,+)$ satisfy $(G,+)\models T$ there is no reason that $(G,+)$ is a cyclic. (For example, by Löwenheim–Skolem theorem there ...