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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Category of $\mathcal{L}$-structures

Given a purely relational language $\mathcal{L}$, the set of $\mathcal{L}$-structures forms a category under the definition of homomorphism found here. Call this category ...
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$T_{\forall }=Th\{\mathcal{M}:\mathcal{M}\hookrightarrow \mathcal{N}\vDash T\}$ [on hold]

1)prove:$T_{\forall }=Th\{\mathcal{M}:\mathcal{M}\hookrightarrow \mathcal{N}\vDash T\}$ answer:$T_{\forall }\subseteq Th\{\mathcal{M}:\mathcal{M}\hookrightarrow \mathcal{N}\vDash T\}$ how to continue? ...
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Proving uncountability of $\mathbb R$ only using the complete ordered field axioms

If we define the real numbers abstractly as a complete ordered field (like described in the Wikipedia page), how can we prove that they are uncountable? In other words, using just the axioms of a ...
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Morley’s Categoricity Theorem for uncountable languages.

Where can I find an accessible exposition of Shelah’s generalization of Morley’s theorem to uncountable languages? (Please, do not answer “Shelah’s Classification Theory”.)
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64 views

Why are structures with no relations called algebras?

"If [a given structure] A has no relations it is termed an algebraic structure, or simply an algebra" - Wolfgang Rautenberg, A Concise Introduction to Mathematical Logic, 3rd edition, page 42. I ...
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Show For any language L two L-structures M and N are elementarily equivalent iff they are elementarily equivalent for every finite sublanguage.

Setting For any language $\mathcal L$, two $\mathcal L$-structures $\mathcal M$ and $\mathcal N$ are elementarily equivalent iff they are elementarily equivalent for every finite sublanguage. ...
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Comparing Category Theory and Model Theory for Master's Thesis.

I am currently doing a Masters thesis in pure maths, and the two current fields that excite me are Category Theory (CT) and Model Theory (MT). I have been reading up on David Marker's Model Theory: ...
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1answer
41 views

Are the implicitly definable sets of a second-order theory the sets the second-order quantifiers range over?

I know that in a second-order setting, due to the failure of the Beth definability theorem, implicit and explicit definition come apart (i.e., there are predicates which can be implicitly, but not ...
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Given L = {<,c0,c1,…} and T3 the theory of DLO with sentence asserting co < c1 < …, Show T4 is complete with four countable models.

Let $\mathcal L_3 = \{<,c_o,c_1,\ldots\}$, and $T_3$ be the theory of DLO with sentences added stating $c_o < c_1 < \ldots$. Now let $\mathcal L_4 = \mathcal L_3 \cup \{P\}$, where $P$ is a ...
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Question 4.5.1 from Marker [closed]

This is question 4.5.1 from Marker. Lifted verbatium but with a correction: Let $\mathcal M = (X,<)$ be a dense linear order, let $A \subset M$ and $\bar b, \bar c \in M^n$ with $b_1 < \ldots ...
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1answer
38 views

Link between definitional expansions and definitional extensions.

I need to prove this, Let $T$ be a theory in language $L$, let $T'$ be a definitional extension of $T$ to language $L\subseteq L'$. If $\mathcal {M} \models T'$, then $\mathcal M$ is a ...
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1answer
56 views

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 ...
3
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50 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 ...
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1answer
42 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 ...
3
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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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70 views

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
58 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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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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29 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 ...
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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. ...
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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
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 ...
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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
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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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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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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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
543 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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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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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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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
63 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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1answer
62 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 ...
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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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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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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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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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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
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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 ...
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1answer
64 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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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
32 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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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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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 ...
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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 ...
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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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56 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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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) ...