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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What are the L-sentences that are true in an empty structure?

I am looking for an algorithm or set of rules to figure out whether a sentence (in first order logic) is true when we are dealing with an empty set as domain. Clearly, it has to be a sentence (no free ...
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19 views

(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 ...
2
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1answer
35 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 ...
2
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0answers
8 views

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 ...
2
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1answer
49 views
+200

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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1answer
34 views

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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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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0answers
26 views

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 ...
2
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1answer
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. ...
3
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0answers
25 views

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
26 views

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 ...
3
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1answer
30 views

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
55 views

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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0answers
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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 ...
2
votes
2answers
63 views

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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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 ...
20
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1answer
537 views

Does there exist any uncountable group , every proper subgroup of which is countable?

Does there exist an uncountable group , every proper subgroup of which is countable ?
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1answer
27 views

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
59 views

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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0answers
40 views

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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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 ...
2
votes
1answer
60 views

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 ...
2
votes
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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2answers
59 views

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
88 views

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 ...
2
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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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35 views

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 ...
3
votes
1answer
33 views

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
votes
1answer
58 views

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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0answers
32 views

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
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 ...
2
votes
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, ...
2
votes
2answers
89 views

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
votes
1answer
41 views

$\kappa$-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
votes
1answer
111 views

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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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. ...
2
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0answers
47 views

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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0answers
46 views

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
19 views

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) ...
3
votes
1answer
39 views

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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0answers
26 views

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 ...
2
votes
1answer
55 views

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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1answer
44 views

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
47 views

Logical Structure of a Proposition

I'm having a hard time figuring out the logical structure of the following theorem : I'm not interested in proving it, for now, i'm just trying to understand its logical structure. I don't know ...
2
votes
1answer
36 views

Independence Property in Model Theory

Often, the Independence Property is defined in the monster model of a complete theory. When it is not, it usually goes like: a formula $\phi(x, y)$ is said to have the independence property if for ...
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2answers
98 views

Definable subsets of $\mathbb{Q}^2$ in $\langle \mathbb{Q} , < \rangle$?

The question seems quite simple; what are the definable subsets of $\mathbb{Q}^2$ over the structure $\langle \mathbb{Q} , < \rangle$. Part of me wants to say there are none, given any definable ...
4
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1answer
81 views

Models of the empty theory

so throughout my reading of model theory the idea of the "empty" theory has been put down as trivial, however I am curious as to why. Let us look at the following. Suppose We have $L_=$, the language ...
2
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0answers
34 views

Algebraic characterization of first order “operations” using limit ultrapowers

In his review http://projecteuclid.org/download/pdf_1/euclid.bams/1183537899 of the Chang and Keisler's classic book "Model Theory," M. Makkai writes: "… let us note that it is possible to formulate ...
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0answers
53 views

Open interpretation of logical theories

This may be more appropriate for MO but I thought I'd ask here first as it's just a question about logic (not my strong point at all but not research-level in itself). I'm going through Razborov's ...
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1answer
73 views

Skolem functions in the real ordered field.

I am currently reading into a bit of model theory and have come across the idea of Skolem functions, as used in the proof of the downward Lowenheim-Skolem theorem. Despite seeing their use there I ...