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Questions tagged [model-theory]

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 languages. Not to be confused with mathematical modeling.

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How is it possible to have a model of a set theory? [duplicate]

I am trying to understand the basics of model theory. Before getting too deeply into it, I would like to know how it is even possible to construct a model, i.e. a structure that satisfies axioms of a ...
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What is the purpose of Semantics/Model theory in Mathematical Foundations?

First off I know very little model theory so apologies if I say anything very dumb or offensive to logicians/model theorists. Second I should note that a lot of what I am saying here is motivated by ...
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Amalgamation base, extending Galois type

Here on the page 12 the Observation 1.11 5) says If $M\leq_{\frak K} N$ are from ${\frak K}_{\lambda}$, $M$ is an amalgamation base and $p\in \mathscr{S}(M) \;\underline{\text{then}}$ there is $q\in ...
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Why is a Skolem theory model complete?

Let $\Delta$ be a Skolem theory. Let $M$ be a model of $\Delta$ and $N$ a substructure of $M$. Then we need to proof that for every $L_N$-sentence $\phi$ we have the equivalence $N \models \phi \iff M ...
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Fixing and pointwise-fixing of a structure in a suqare

Suppose we have this square in the context of Abstract Elementary Class ${\frak K}=(K,\leq_\frak K)$: $$ \require{AMScd} \begin{CD} N_1 @>f_1>> N_3 \\ @A{\preceq_\frak K}AA @AA{f_2}A\\ M_0 @&...
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1answer
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Ultraproduct of simple structures

Definition. 1) A complete first-order $\mathcal{L}$-theory $T$ is said to be simple if each type does not fork over some subset $A$ of its domain where $|A|\leq |T|$. An $\mathcal{L}$-structure $\...
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Question about $\Gamma_\infty^M$ deriving a very specific wff

In p.69 of Tourlakis's mathematical logic book. He pulls a very specific theorem seemingly out of thin air. $\Gamma_\infty^M \vdash \exists_x(f \bar n ... = x)$ I'm not sure how this is derived as ...
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Elementary equivalence of standard and non-standard model of arithmetic

There is the common construction of a non-standard model of arithmetic by adding a constant symbol c to the signature and adding $\{n<c:n\in \mathbb{N}\}$ to the theory PA. Now by adding all ...
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Base enlargement/isomorphism of models over a model [closed]

How can I see that if models $M_0$ and $M_1$ are isomorphic over $N_0$ and $N_0\leq_{\frak K} N_1$ then they are isomorphic over $N_1$? It is almost obvious but not quite.Any simple argument will help....
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Why does the compactness theorem not apply to infinite languages?

I am trying to show that compactness doesn't apply to infinite languages like $L_{\omega_1,\omega}$ that allow infinite FOL sentences like $\forall x\, \bigvee_{n\in \omega} x \approx S^n(0)$, i.e. $\...
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1answer
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Proving that $\langle\mathbb{R}-\{7\},<\rangle$ is an elementary substructure of $\langle\mathbb{R},<\rangle$

Prove that $\langle\mathbb{R}-\{7\},<\rangle$ is an elementary substructure of $\langle\mathbb{R},<\rangle$. What I thought I should do is to use induction on statements to prove that. Any ...
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1answer
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What does this set-theoretic notation mean in the proof of this theorem?

From A friendly introduction to mathematical logic, In the first red box below, what does this notation mean? And in the second red box, how exactly is this process repeated?
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Can we have models that do not satisfy unproved existential sentences in the theories they model?

For any first order theory $T$, let's say that $M$ is an ideal existential model of $T$ if and only if, for every formula $\varphi(x)$ in the language of $T$ in which only the symbol "$x$" appears ...
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The theory of dense linear orders without end-points is not $2^\omega$-categorical

It seems best to prove this by counter example. Both $\mathbb{R}$ and $I := \mathbb{R} \backslash \mathbb{Q}$ under the usual order $<$ are models of the theory of dense linear orders without end-...
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Prove that all algebraic numbers are included in any elementary substructure of $\mathbb R$

Let $A$ be an elementary substructure of $\mathbb R$ where $\mathbb R$ is $\langle \mathbb R,+,\cdot,0,1\rangle$ . Show that $A$ contains any algebraic number. What I tried to do was use the fact ...
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1answer
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Reduct of a first-order structure

I am trying to understand the notion of reduct in model theory. I found the following definition in Hodges's model theory book. Definition. Let $\mathcal{L}$ be a first-order language and $\mathcal{L}...
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How to give formal proof in logic?

I am taking a course on mathematical logic and am struggling to give formal proofs of theorems or claims. Currently doing first order logic. Here, an example: Claim: $\models (\exists x)\big(A(x)\...
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Why is it that the complete L-theories containing $\Sigma$ are exactly those of models of $\Sigma$?

Consider a set of L-sentences $\Sigma$ and a set of complete L-theories $T_i$ containing $\Sigma$ i.e. $\Sigma \subseteq T_i$. Why is it that for each complete L-theory $T_i$ containing $\Sigma$ are ...
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1answer
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Criterion for $\forall\exists$-axiomatizability.

I am stuck in following exercise: Let $T$ be a theory with following property: for all models $M$, $N$, and $P$ of $T$ if $M \subset P$, $N \subset P$, and $M \cap N \neq \emptyset$, then $M \cap N ...
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how a theory can be categorical of a large cardinality?

ehrenfeucht mostowski models eliminate types,also we can find saturated models of arbitrary large cardinalities, so I got confused how a theory can be categorical of that large cardinality?
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Boolean algebras/Unknown notation

Does someone know what is meant (in the context of trees and Boolean algebras by Shelah) here on the page 8 right above Remark 1.5: $$\{\langle\rangle\}\cup\{\langle\xi\rangle\otimes_{\zeta(*)}d\eta:\...
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Chang & Keisler Exercise 4.3.9

I'm having a lot of trouble sorting out an exercise from Chang & Keisler's Model Theory: 4.3.9: Let $I$ be an infinite set of power $\alpha$. If $E \subset P(I)$, $|E| \leq \alpha$, and the ...
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1answer
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Conjunction closure of positive primitive formulas

Let $\mathfrak M = (M, 0, +, - , r)_{r\in R}$ be a structure which realizes a module, i.e., $(M, 0, +, -)$ is an abelian group together with operations $r : M \to M$ with $R$ a ring with $1$. Let $L_{...
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Intuitive understanding of definitional expansion in model theory

I was trying to understand what definitional expansion mean. Consider having two L-structures $\mathcal A$ and $\mathcal A'$ where $A'$ is an L'-expansion of $\mathcal A$. We way that $\mathcal A'$ is ...
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Why is it if every model of $\Sigma$ has an L'-expansion then $\Sigma'$ is conservative over $\Sigma$?

Consider $\Sigma \subseteq \Sigma'$ and $L \subseteq L'$. Then the theorem is: If every model of $\Sigma$ has an L'-expansion, then $\Sigma'$ is conservative over $\Sigma$. Recall conservative ...
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Natural models of Ackermann Set Theory

Consider Ackermann's Set Theory (as described here, §6, including the axiom of regularity for sets), henceforth denoted AST, as a theory in the language $\left\langle \in,\mathbf{V} \right\rangle$ ...
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The cornerstone definition in Abstract Elementary Classes

In the first paragraph of p.41 of Introduction to: classification theory for abstract elementary class, Shelah gives the following definition of (Galois) type. For $M\preceq N_\ell$ and $a_\ell\in ...
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Q: On Soundness proof in Tourlakis 2003

In p.59 of Lectures in Logic and Set Theory, Tourlakis proves the inductive step for the E-introduction rule of Soundness. He uses a proof by contradiction stating that $B'[x \leftarrow \bar i] \...
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1answer
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Sentence over the empty language saying a model is infinite

So the question is whether there is a sentence $\phi$ over the empty language such that for any structure $M$, $M\models\phi$ iff $M$ is infinite. I’m pretty sure the answer is no, but I don’t see ...
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A theory $T$ is model-complete if the union of $T$ with an atomic diagram is complete

Let $T$ be a theory in first order logic over some language $L$. Let $\mathfrak A$ be some structure over $L$ with $\mathfrak A \models T$ and with $A$ be its universe. Then consider every $a \in A$ ...
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What is the relationship between the common concept of “model” and “model” as used in Model Theory? [duplicate]

To my understanding, a model in Model Theory is an interpretation (in a form of a set or other algebraic structures) for a certain sentence S which makes S true. In everyday language, and also in ...
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1answer
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Notion of submodel relation

There is no definition of the essential notion of substructure (=submodel) in Shelah's introduction E56 to AEC, 1st Volume. Could someone please define this for me? I think that $$M \subseteq N$$ ($M$ ...
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The structure $(\mathbb N, <, f)_{f\in \mathcal F}$ seems to have no elementary extension

This is exercise 2.3.1 (2) from Tent/Ziegler, A Course in Model Theory. Let $\mathcal F$ be the set of all functions $\mathbb N \to \mathbb N$. Show that $(\mathbb N, <, f)_{f\in \mathcal F}$ ...
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1answer
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Find non-isomorphic models of $(Q,<,c_{n \in N})$

This is a problem in Basic Model Theory by Kees Doets: Let $X=(Q,<,n)_{n \in N}$ $Y=(Q,<,\frac{-1}{n+1})_{n \in N}$ $Z=(Q,<,q_n)_{n \in N}$ where $\{q_n\}_{n \in N}$ is an ascending ...
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1answer
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Existence of saturated models of a theory.

In Tent, Ziegler: A Course in Model Theory it is stated on page 89, that Saturated structures need not exist (think about why not), but by considering special models instead, we can preserve many ...
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1answer
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Isomorphic subsets of countable total orders

Suppose $(\Omega,\leq)$ is a totally ordered set, with $\Omega$ infinite and countable. If $S$ is an infinite subset of $\Omega$, then $(S,\leq)$ denotes the induced totally ordered set. Are there ...
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Does naturality imply independence from choice?

My question concerns the notion of naturality defined in Section 12.3 of Hodge's (longer) Model Theory. See the addenda for the definition. Hodges proves a result that if an algebraic construction ...
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1answer
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Let $L = \{<\}$ prove there is no $L$-theory that has precisely the well ordered sets as models

This is the literal question: let $L$ be the language that only contains the two place relation symbol $<$. Prove there is no $L$-theory that has precisely the well ordered sets as models. I ...
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What do sentences in the theory of the structure $A=(Q,<,n)_{n \in N}$ look like?

I'm working on a problem from Kees Doets, and he mentions the following structures: $X=(Q,<,n)_{n \in N}$ $Y=(Q,<,\frac{-1}{n+1})_{n \in N}$ $Z=(Q,<,q_n)_{n \in N}$ where $\{q_n\}_{n \in ...
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1answer
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model theory - existence of a model

Let $\Gamma$ be a consistent set of $L$-sentences with infinite cardinality. If it has an infinite model, then there exists a model for $$ \Gamma' =\Gamma \cup \{\lnot c_a = c_b : a \neq b \}, $$ ...
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If $\Sigma \models \phi$, then for some finite $\Delta \subset\Sigma$, $\Delta \models \phi$.

This is an easy consequence (Doets calls it Compactness Theorem (version 2)) in Kees Doets' Basic Model theory: Let $\Sigma$ a set of sentences and $\phi$ a sentence. If $\Sigma \models \phi$, then ...
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Boolean algebras/ Shelah/ Unclear step in the proof

Here on the page $10$, what does the displayed formula in the $6$th line $$\text {lg}(\eta_\ell)<\omega\Rightarrow \bigcup\{\text{Rang}(\nu_\ell(k):k<\text{lg}(\nu_\ell))\}\cap\bigcup_{k<\...
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1answer
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model theory - completeness of a theory

For an $L$-structure $\mathcal{A}$, the $L$-theory of $\mathcal{A}$ is the set of $L$-sentences: $$ \mathrm{Th}_L(\mathcal A) = \{\sigma : \mathcal A \models \sigma\} $$ Prove that $\mathrm{Th}...
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2answers
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Craig interpolants for inifinite set of implications

Suppose we have an infinite set of first-order sentences $\{\alpha_i\}_{i=1}^{\omega}$ and first-order sentence $\beta$ such that for all $i$, \begin{align*} \alpha_i \vDash \beta \end{align*} I ...
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1answer
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What is the difference between the logical formula $\alpha$ and expression “$\alpha$ is true”?

What is the difference between the logical formula $\alpha$ and expression "$\alpha$ is true"? I feel that there is a difference between those expressions. I think this difference is the difference ...
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1answer
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Ultraproducts with a non-principal ultrafilter is aleph-one compact

I came across the following statement: Given an $\omega$-sequence of $\mathcal{L}$-structures, $(\mathcal{M}_i : i < \omega)$, and a non-principal ultrafilter $\mathcal{U}$ on $\omega$, the ...
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1answer
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Prove that the topological closure of a set is definable if the set is definable

Let $L$ be the language $L=\{<,=,+,-,\cdot, 0,1\}$, with standard interpretations, and let $\mathcal{A}=\langle\mathbb{R}, <,=,+,-,\cdot,0,1\rangle$. Let $S\subseteq\mathbb{R}^n$. Show that if $...
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0answers
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A model of $PA^\omega$ in ZFC [closed]

It is known that the existence of a model (i.e. consistency) of $PA$ is provable in ZFC, the set $\omega$ equipped with usual operation on natural numbers is indeed a model of $PA$, and we can think ...
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2answers
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Logic problem about set of disjunction forms(Fixed)

I will call the disjunction of literals a disjunction form. Let $\Sigma$ be a set of disjunction forms, and $\alpha$ is a well-formed formula satisfying $\Sigma \vDash \alpha$. For all propositional ...
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1answer
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Dimension of vector spaces under surjection

Definition. Let $V$ be a vector space and $X\subseteq V$ be a definable subset in the language of vector spaces. $X$ is independent if for all $a\in X$, $a\notin span(A\setminus\{a\})$. Definition. ...