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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Uniqueness of an embedding theorem for Real differential fields

I will follow a preliminary exposition for the problem in question, which will essentially follow the format on http://www4.ncsu.edu/~singer/papers/model_diff_fields.pdf [pg. 87]: Let $K$ be a real ...
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Robinson's Consistency Theorem for first order languages

Is there a simple proof for the case of first order languages for this theorem? Let $L_1$,$L_2$ be first order languages and $L$=$L_1$ $\cap$ $L_2$. Let $T_1$, $T_2$ be consistent ...
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“Adding constant symbols” in Model Theory.

What is going on when we "add constant symbols" to extend a language L. A new constant is not in the "alphabet" of L. C is, for example, just { 0, 1 } in many cases, but C could be an infinite set. ...
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46 views

Model complete theories without quantifier elimination

As we know, if a theory $T$ admits quantifier elimination, then $T$ is model complete. What are the simplest examples that show that the converse is not true?
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What does not having a first-order frame imply for models?

There are formulas in modal logic which which do not have a first-order frame condition, as stated here (Non-Sahlqvist formulas, Wikipedia). An example is the McKinsey formula for $p$: ...
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For any propostional sentences $a,b,c$, if $a\models (b\wedge c)$, then $a\models b$ and $a\models c$

I'm having a hard time dealing with the $\models$ symbol. I don't really know how to reason through or manipulate these equations to prove why or why not the result holds. Another similar question is: ...
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Motivation behind automorphism bases?

Given a model $\mathcal{M}$ with a domain $M$ and $B \subseteq M$, $B$ is an automorphism base for $\mathcal{M}$ iff $\forall f \in Aut(\mathcal{M}). (\forall b \in B. f(b)=b) \implies f = ...
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1answer
43 views

Model of Robinson Arithmetic but not Peano Arithmetic

I am curious how can structure of with linear successor function that do not admin induction looks like. In other words I would like to see structure of Peano Arithmetic without induction. I believe ...
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1answer
81 views

The Lowenheim-Skolem theorem does not hold for $\mathfrak{L}_{II}$.

In "Mathematical Logic" second edition written by H-D Ebbinghaus, J.Flum and W.Thomas, in chapter 9 "Extensions of First-Order Logic", page 140, in the prooof of theorem 1.5 (The Lowenheim-Skolem ...
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If $B$ is a model for the set of positive consequences of $\Gamma$, then there's $A \subseteq B$ such that $A \models \Gamma$

I'm working through Chang & Keisler again and got stuck on the following exercise (1.2.14) about propositional logic. First, consider a set $\mathscr{S}$ of sentence symbols of arbitrary ...
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1answer
30 views

Universal theory with relation symbol

Let $\mathcal{L}$ be a language containing a binary relation symbol $R$. I need to prove that if $T$ is a universal $\mathcal{L}$-theory (by which I mean that $T$ is a collection of universal ...
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57 views

Does theory have the smallest model

I have a language $L=\{P\}$ with equation, where $P$ is binary predicate symbol. Language's formulas are: $\varphi \equiv \forall x \forall y (\neg P(x,x) \land (P(x,y) \to P(y,x)))$, $\psi \equiv ...
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1answer
37 views

Does theory have uncountably many pairwise non-isomorphic models?

I have a language $L=\{P\}$ with equation, where $P$ is binary predicate symbol. Language's formulas are: $\varphi \equiv \forall x \forall y (\neg P(x,x) \land (P(x,y) \to P(y,x)))$, $\psi \equiv ...
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1answer
48 views

Equivalence vs equisatisfiability

Wikipedia page states that first order formula after skolemization is equisatisfiable but not equivalent to original one. I do not understand what the difference is. I know definition of ...
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In plain language, what's the difference between two things that are 'equivalent', 'equal', and 'identical'?

In plain language, what's the difference between two things that are 'equivalent', 'equal', 'identical', and isomorphic? If the answer depends on the area of mathematics, then please take the ...
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1answer
43 views

Find some complete theory $U \supseteq T$

I have a language $L=\{P\}$ with equation, where $P$ is binary predicate symbol. Language's formulas are: $\varphi \equiv \forall x \forall y (\neg P(x,x) \land (P(x,y) \to P(y,x)))$, $\psi \equiv ...
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1answer
46 views

Problem with forking

Let $A \subseteq \mathbb{U}$ be a small set, and $\bar{a}= (a_1, \dots, a_n)$, $\bar{b}=(b_1, \dots, b_k)$ be tuples of $\mathbb(U)$. Show that tp$(\bar{a}/A\cup (b_1, \dots, b_k)$ does not fork over ...
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1answer
50 views

Morley rank (with an unusual definition)

For a definable set $X \subseteq \mathbb{U}^n$, let us denote $\text{RM}(X)$ the Morley Rank $\text{RM}(\varphi(\bar{x}))$, with $\varphi(\bar{x})$ the formula defining $X$. Show that, for $X ...
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1answer
36 views

Does an interpretation of a structure by itself induce a bijection on the automorphism group of the structure?

Let $\Gamma$ be a model-theoretic interpretation of a structure $B$ in a structure $A$. Then $\Gamma$ induces a group homomorphism $\alpha_\Gamma:\mathrm{Aut}(A) \rightarrow \mathrm{Aut}(B)$. (See, ...
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67 views

Prove the Robinson arithmetic has infinite non-isomorphic models

I found this question: Can finite theory have only infinite models?, where is proved that Robinson's Arithmetic can have infinite models, but I've been unable to prove or find a proof of the existence ...
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Consequence in $\mathcal{L}_{\infty\lambda}$

Consider the infinitary first-order language $\mathcal{L}_{\infty\lambda^+}$ whose non-logical vocabulary consists of $\lambda \geq \omega$ individual constants and countably many predicate constants ...
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1answer
29 views

How would one prove that satisfaction of closed formulas is valuation-independent? (In FOL)

Consider this proposition in first-order logic: For any interpretation $I$, any closed formula $\phi$ and any two valuations $\rho$, $\sigma$. $I\rho \models \phi \iff I\sigma \models \phi$ This is ...
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1answer
35 views

Countable transitive model of ZFC and $\mathcal{P}(\omega)$

Let $\mathbb{M}$ be a countable transitive model of ZFC. I understand that $\omega^\mathbb{M} = \omega$ but $\mathcal{P}(\omega)^\mathbb{M} \neq \mathcal{P}(\omega)$, for ...
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Model Theory - Equivalence of formulas using automorphisms

Let $\mathbf Q$ denote the additive group of rational numbers, i.e. the structure $\mathbf Q = (\mathbb Q;+,0)$. Let $L$ be the language of $\mathbf Q$ and let $T$ be the complete theory of $\mathbf ...
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1answer
42 views

Why are $n$-ary function symbols interpreted as $n$-ary operations and not general $n$-ary functions?

I would appreciate if someone here could check the correctness of my reasoning about the following. I'm new to logic. Let $\mathfrak{M} = \langle M,\mathfrak{I} \rangle$ be a structure for some ...
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2answers
52 views

Ultraproducts and Elementary Embeddings

Let $K= \{A_i: i\in \omega\}$ be a countable collection of $L-$structures. Suppose that for each $A_i, A_j$ in $K$, $\exists A_p \in K$ such that $i,j< p$ and $A_i \prec A_p $ and $A_j \prec ...
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51 views

Is this an accurate description of structures and interpretations.

I read about structures and interpretations today. I've described them below this paragraph. Have I accurately described them? If not, what have I incorrectly described? A structure, $\mathscr{A}$, ...
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1answer
47 views

Let $T$ be the theory of linear, dense order, without minimum or maximum. Is $T\cup\{c_{i}<c_{j}\mid i<j\}$ complete?

Let $T$ be the theory of linear, dense order, without minimum or maximum in the language $\mathscr{L}$ . Expend the language by adding it countable amount of constants: ...
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28 views

Quick question about the relation between elementary classes and pseudo-elementary classes

Let $\mathcal{L}$ be a logic and $\mathscr{K}$ a class of structures in the vocabulary of $\mathcal{L}$. We say that $\mathscr{K}$ is a (basic) elementary class iff there is $\phi \in \mathcal{L}$ ...
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65 views

$T$ be a theory that claims that both $P$ and $\neg P$ are infinite. Prove that $T$ is a complete theory

Let $\mathscr{L}=\{P\}$ a language with one unary predicate. Let $T$ be a theory that claims that both $P$ and $\neg P$ are infinite. Prove that $T$ is a complete theory. I tried to prove by ...
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2answers
140 views

Unexpressibility of a property in first order logic

We can give a very general notion of what is to iterate a function. Given a set $\mathcal U$ and a function $f:\mathcal U \rightarrow \mathcal U$, then, to iterate the function $f$ will mean to ...
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1answer
35 views

Models of $T_\exists$ are precisely the structures that contain a model of $T$

An usual exercise in introductions to model theory is to prove the following statement: Let $T$ be a theory, and let $T_\forall$ be the set of all universal sentences that are consequences of $T$. ...
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Ordering of $\mathbb{R}$ not quantifier-free definable in $L_{R}$

I'm reading David Marker's book "Model Theory: An Introduction" and I'm trying to solve Exercise 3.4.24 which is stated as follows: Let $x$ and $y$ be algebraically independent over $\mathbb{R}$. a) ...
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2answers
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Formula for perfect squares spectrum.

I have been working on exercises from "A first Course in Logic" by S. Hedman. Exercise 2.3 (d) asks to find a first-order sentence $\varphi$ having the set of perfect squares as a finite spectrum. But ...
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29 views

Models of $T$ in cardinallity $\kappa$ are isomorphic

Assume that $T$ is a consistent set in a countable language $\mathscr{L}$ with no finite models. There is a cardinal $\kappa$ such that every two models of $T$ with cardinallity of $\kappa$ are ...
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Are there number systems corresponding to higher cardinalities than the real numbers?

As most of you know, the set $\omega$ with cardinality $\aleph_0$ corresponds to what we normally know as the natural numbers $\mathbb{N}$, and the set $\mathcal{P}(\omega)$ with cardinality ...
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67 views

Is this theory complete?

I have a language $L=\{P\}$ with equation, where $P$ is binary predicate symbol. Language's formulas are: $\varphi \equiv \forall x \forall y (\neg P(x,x) \land (P(x,y) \to P(y,x)))$, $\psi \equiv ...
0
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2answers
74 views

Adding Substructures by Forcing

Consider a first order language $\mathcal{L}$ and an $\mathcal{L}$-structure $M$. Let $V$ be a model of ZFC (or ZF) the general question is that what would happen to classes, $Sub(M):=\{N~;~N~\text{is ...
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1answer
39 views

Equivalence Relation on a Class

I'm trying to understand Scott's proof of the incompatibility of axiom of constructibility and the existence of a measurable cardinal. I'm stuck in the use of Łoś's Theorem in the universe. Jech's ...
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1answer
75 views

Why are models in logic called models?

A model is an interpretation of a given formal language under which any wff in a given set of wffs of this formal language is true. Why are models called models? What's the reasoning behind the name? ...
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Is $F(x_1,x_2,\dots,x_n)$ where $(x_1,x_2,\dots,x_n)\in \Delta$,a semi-algebraic function?

Given $$F(t_1,t_2,\dots,t_n)=\int\frac{P_1(x_1,x_2,\dots,x_n)}{P_2(x_1,x_2,\dots,x_n)}dx_1dx_2\dots dx_n$$ where $P_1(x_1,x_2,\dots,x_n), P_2(x_1,x_2,\dots,x_n)$ are polynomials whose coefficients ...
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Uncountable reals in the theory

The Question I'm looking for a possibility to somehow proof the "essence" of Cantor's diagonal argument within a recursive first-order theory which is satisfied by the reals (better: within a theory ...
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38 views

A universal formula is not equivalent to an existential formula

Suppose a formula is looks like the following: $\forall x_1 ... \forall x_n \alpha$ Where $\alpha$ is a formula free of quantifiers. And if $P$ is a 1-ary relation letter, then the formula ...
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2answers
212 views

Can it be decidable for any polynomials to have the intersecting point?

Give system of polynomials$$P_1(x_1,x_2,\dots,x_n)=0,$$$$\vdots,$$$$P_k(x_1,x_2,\dots,x_n)=0$$ Can it be decidable for thoses polynomials to have the intersecting point ?
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1answer
36 views

Algebraic invariants for first order equivalence between fields

I know that every two models of the theory $ACF$ (namely two algebraic closed fields) with the same characteristic are elementary equivalent. But what about generic fields? Are there any algebraic ...
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28 views

Types linearly-ordered by deduction?

I'm wondering whether anyone has come across the following concept before: Consider a first-order language $L$ and a type $p$ over a theory $T$. I say that $p$ has a well-ordered filter-base if there ...
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Uncountably Categorical Theories and Embeddings

Suppose that $T$ is uncountably categorical. By the Baldwin-Lachlan Theorem, we note that $I(T,\aleph_0)=1$ or $\aleph_0$. Suppose that $I(T,\aleph_0)=\aleph_0$. Is it always the case that we get ...
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1answer
32 views

Use of model theory in flag algebras

I need to learn about Razborov's "flag algebras" (see http://bit.ly/1u1a1NB) to solve a problem about graphs. Flag algebras are a very general new algebraic tool for studying combinatorial structures. ...
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75 views

Is there a rationality-preserving order isomorphism between $\mathbb{Q}$ and two disjoint open intervals?

I have a homework question in a intro logic course, part of which requires me to Find an order preserving isomorphism between $\mathbb{Q}$ and $\mathbb{Q} \cap ((0,1) \cup (2,3))$. So, I need an ...
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1answer
42 views

Model-theoretic question about language of field theory.

Let $\mathscr{L}=\{+,·\}$ be the language of the theory of fields. Let $\phi$ be a sentence in this language. Show, using the compactness theorem of first-order logic, that if $\phi$ holds in finite ...