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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Is equality axiomatizable in first order logic without equality?

In the language of first order logic without equality but with a single binary relation symbol, is the class of equality relations an axiomatizable class?
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30 views

What is the stone space $S_n(T)$ for a theory with infinitely many equivalence classes, each class infinite? [on hold]

What is the stone space $S_n(T)$ for each $n$, for a theory $T$ with infinitely many equivalence classes, each class infinite.
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27 views

Proof verification for structure construction

This question is from Enderton's mathematical logic. Question 8 sec 2.5 pg 146. It says assume the language that has $\forall$ and P, where P is a two place predicate symbol. Let A be the structure ...
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39 views

Is there a 'largest' second-order categorical axiomatization of set theory, extended from ZFC2

While it's possible to obtain categorical second-order axiomatizations of set theory by extending ZFC2 with additional axioms (see [1]), these axioms seem somewhat arbitrary (e.g. adding an axiom that ...
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40 views

K is finitely definable if it has a finite support

I tried to prove that, but without a succes: Prove that K is finitely definable if and only if it has a finite support. *support of a set of assignments K is a set S that contains the atomic ...
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0answers
32 views

Union of definable sets is a definable set [duplicate]

I tried to prove this question but without a success: Let $K_1 \text{and } K_2$ be definable sets, prove that $K_1∪K_2$ is definable. What I tried to do is to assume: $K_1=\text{Ass}(X)=\{v\mid ...
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1answer
57 views

Exercise $ 3.4.15 $ of David Marker’s “Model Theory”.

I was reading David Marker’s Model Theory and came upon the following problem in Chapter 3. Setting Let $ \mathcal{M} $ be a saturated $ \mathcal{L} $-structure. A definable subset $ X \subseteq M ...
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2answers
109 views

Union of definable sets

I tried to prove this question but without a success: Let $K_1$ and $K_2$ be definable sets, prove that $K_1\cup K_2$ is definable. What I tried to do is to assume: $K_1=Ass(X)=\{ v|v \vDash X \}$ ...
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31 views

Examples and applications of homogeneus models in model theory.

Does anyone know any specific examples or applications of homogeneus models, to model theory or any other branch? For example, an application would be that prime models are isomorphic in a countable ...
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67 views

How can the axioms (and primitives) of Tarski's axiomatization of $\Bbb R$ be independent?

While reading through this Wikipedia page about Tarski's axiomatization of the reals, a particular bit of text jumped out at me: Tarski proved these 8 axioms and 4 primitive notions independent. ...
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1answer
27 views

Sequence of indiscernibles in a theory with an equivalence relation with infinitely many equivalence classes

Let $\mathcal L$ be a language with a single binary relation $E$, and the theory $T$ where $E$ an equivalence relation with infinitely many equivalence classes, each of which is infinite. Are its ...
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1answer
41 views

Show that if $T$ is not $\aleph_0$-categorical then $T$ has a non-atomic model of size $\aleph_1$

Exactly as the title stated: Show that if $T$ is not $\aleph_0$-categorical then $T$ has a non-atomic model of size $\aleph_1$ Would like some pointers on how to proceed.
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1answer
47 views

What is satisfiable by a reduct of a model is satisfiable by the original model (and vice versa)?

My professor told me that any formula that is satisfiable by a reduct of a model is satisfiable by the model it is a reduct of, and vice versa (as long as the formula is interpretable on the ...
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1answer
43 views

What is the theorem that shows that second-order logic is the ceiling of model characterization?

I was reading this blog posting and the following claim was made: ...[T]here's tricks for making second-order logic encode any proposition in third-order logic and so on. If there's a collection ...
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38 views

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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19 views

$T_{\forall }=Th\{\mathcal{M}:\mathcal{M}\hookrightarrow \mathcal{N}\vDash T\}$ [closed]

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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3answers
49 views

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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32 views

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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1answer
65 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 ...
2
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1answer
48 views

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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2answers
88 views

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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1answer
45 views

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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1answer
39 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
58 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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53 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
43 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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35 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 ...
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2answers
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
61 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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65 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 ...
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. ...
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27 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
28 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
57 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 ...
2
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2answers
68 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
34 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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1answer
560 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
61 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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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
68 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
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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 ...
2
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2answers
39 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
63 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 ...
1
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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 ...
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1answer
35 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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1answer
35 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 ...