Model theory is the branch of mathematical logic which deals with the connection between a formal language and its interpretations, or models.

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Real closed field+ model thoery

Is it true that every real closed field can be elementarily embedded in some other real closed filed with the same Archimedean classes (I mean in a proper extension)? Can for example real numbers be ...
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Example of two structures

This is probably a very trivial question, still I don't seem to find an answer. I'd like to see an example (in some language) of two countable structures $\mathcal{M}_1 $ and $ \mathcal{M}_2 $ with ...
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Determining the exceptional set in the theorem of Ax & Kochen

Ax & Kochen [1] proved that for every $d\in\mathbb{N}$ there exists a finite set $A(d)$ such that for every prime $p\not\in A(d),$ every homogeneous polynomial of degree $d$ over $\mathbb{Q}_p$ in ...
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Is the set of undecidable problems decidable?

I would like to know if the set of undecidable problems (within ZFC or other standard system of axioms) is decidable (in the same sense of decidable). Thanks in advance, and I apologize if the ...
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148 views

Interdefinability of two expansions of the Real Field

I was asked the following question two days ago, but I couldn't completely resolve it. Here is the claim: $\mathcal R = (\mathbb R,+,\cdot)$ is the real field. Let $I$ be an open interval (perhaps ...
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On the theory of infinite extraspecial $p$-groups

$p$ is a prime number. A group $G$ is called an infinite extraspecial $p$-group if 1) it is infinite, 2) every $g\neq 1$ in $G$ has order $p$, 3) its centre $Z(G)$ coincide with $G'$ and is a ...
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Intersecting the algebraic closure of independent elements

$G$ is a group with a simple first order theory $T$ as defined by Shelah, hence equiped with a "nice" notion of independence. $G$ also has generic elements. I write $acl^n(A)$ for the set of elements ...
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are there standard examples of stable theories that are undecidable?

What is known about decidability of various first order theories studied in stable model theory, geometric model theory, o-minimality? For example, is there a natural example of an undecidable ...
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Connes' embedding conjecture for uncountable groups

In this topic, I will use the word uncountable group referring to groups whose cardinality is $\leq|\mathbb R|$. Notation: $R$ is the hyperfinite $II_1$-factor, $\omega$ is a free ultrafilter on the ...
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682 views

A Model-Theoretic Helly's Theorem

There is a combinatorial question posed to me (or rather, posed near me) by my adviser. I am having quite a lot of difficulty proving it. It goes: For any NIP theory $T$ (complete with infinite ...
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359 views

Showing that every satisfiable sentence with at most two variables has a finite model

I have tried to prove, in first order logic, that every satisfiable sentence (without function symbols) with at most two variables has a finite model. My attempts were unsuccessful. This is an ...
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394 views

Examples of Graphs with Trivial Definable Closure

If I'm giving a lecture on trivial definable closure (as a property of graphs), what is a good example of a graph that I can easily draw to depict the concept of trivial dcl?
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386 views

Definable subsets of the integers as an abelian subgroup?

Consider the integers as a first-order structure in the language {0,+,-} of abelian groups. I suspect that the collection of definable subsets (without parameters) of this structure is an algebra ...
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267 views

Extending elementary maps in stable theories.

Take some stable theory $T$ with elimination of imaginaries, all sets appearing are small subsets of the monster model of $T$, elementary maps are restrictions of automorphism of the monster model of ...
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999 views

Definable measure preserving isomorphisms of $p$-adic semialgebraic sets

Hi, Consider a $p$-adic field $K$ (finite extension $\DeclareMathOperator{\bQ}{\mathbb{Q}}$of $\bQ_p$) in Macintyre language $\DeclareMathOperator{\cL}{\mathcal{L}}$ $\cL_{\rm Mac}$. Let $Z$ be a ...
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New results on Chow's notion of closed-form numbers?

In an interesting article (available here), Timothy Chow proposes that a closed-form number be defined as an element of the smallest subfield of $\mathbb{C}$ that is closed under $\exp$ and a chosen ...
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6answers
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A book about model theory

I am looking for a good book about model theory. As this is obviously too vague, let me explain what I am looking for and why. First I am interested about the basics and foundations of model theory. ...
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287 views

“Duals” of Lindenbaum algebras

From Wikipedia I learn: The Lindenbaum algebra A of a theory T consists of the equivalence classes of sentences of T. The operations in A are inherited from those in T. If there are ...
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Are all countable, nonstandard models of arithmetic given by ultrapowers?

Countable models of PA fall into two categories: the standard one $(\omega, S)$ and the nonstandard ones (all the rest). The only way I've seen to construct a nonstandard model is through taking an ...
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What's a magical theorem in logic?

Some theorems are magical: their hypotheses are easy to meet, and when invoked (as lemmas) in the midst of an otherwise routine proof, they deliver the desired conclusion more or less ...
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331 views

Stable theory: question about definability of independece

I would like to know why the relation: R(x,y) iff x independent from y (i.e: tp(x/y) doesn't fork over the empty set) is type definable in stable theory? Thanks to the helper.
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391 views

Ergodic Invariant Measures and the Rado graph

Given two or more invariant measures on a structure, there are various ways to combine them to form another invariant measure on the structure. For example, given two invariant measures on a ...
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442 views

Generalizations of PA and its standard and non-standard models

Consider Peano's axioms — in its first-order version and without addition and multiplication — with its single injective function $S$: $(\forall x) \neg Sx = 0$ $\Big(\phi(0)\ \ \&\ ...
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What are some other uses for Ehrenfeucht-Fraïssé games?

Let $\mathfrak{A} = \langle A, \dots \rangle$ and $\mathfrak{B} = \langle B, \dots \rangle$ be structures for a signature $\mathscr{L}$. For each ordinal $\gamma$ we define a game of perfect ...
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660 views

ZFC, set membership and FOL

Hi, Is set membership defined in the signature of ZFC, or is it *specified" in the signature of ZFC? The wikipedia article http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory says that ...
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Categoricity in second order logic

Hi, It's shown by an easy cardinality argument that there are complete second-order theories that are not categorical (have more than one model up to isomorphism). Anyone knows of a concrete example ...
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655 views

What is the spectrum of possible cofinality types for cuts in an ordered field? Or in a model of the hyperreals? Or in a nonstandard model of arithmetic?

I am interested to know the full range of possibilities for the cofinality type of cuts in an ordered field and in other structures, such as nonstandard models of arithmetic. Definitions. ...
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658 views

Is $Ded(\kappa)<Ded(\kappa)^\omega$ consistent?

Hello, I want to ask if anyone can tells us what is known (consistently) about $Ded(\kappa)$, $\kappa$ an infinite cardinal. Definition If there is a dense linear order w/o endpoints of size ...
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What are some proofs of Godel's Theorem which are *essentially different* from the original proof?

I am looking for examples of proofs of Godel's (First) Incompleteness Theorem which are essentially different from (Rosser's improvement of) Godel's original proof. This is partly inspired by ...
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721 views

Cherlin's “Main Conjecture”

Cherlin's "Main Conjecture" from his 1979 paper "Groups of Small Morley Rank" is the following: Every simple $\omega$-stable group is an algebraic group over an algebraically closed field. Zilber was ...
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224 views

Reference wanted for the theory of pseudofinite models

If $\mathcal L$ is a first order language and $\mathcal T$ is theory over $\mathcal L$, then a model $\mathcal M$ of $\mathcal T$ is pseudofinite if it satisfies all sentences satisfied by all finite ...
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Pointwise algebraic models of set theory

Let $\mathfrak{M} = \langle M, E \rangle$ be a structure for the language of set theory, and take some $B \subseteq M$ and $m \in M$. Say that $m$ is definable over $B$ iff there is a formula ...
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Is there a relational countable ultra-homogeneous structure whose countable substructures do not have the amalgamation property?

The question can be stated in a fashion not requiring much background: Let $M$ be a countable ultra-homogeneous relational structure - namely, a countable set equipped with a bunch of relations on ...
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568 views

truth vs. provability for ordered fields

In Propositions equivalent to the completeness of the real numbers I started by asking "Can anyone point me to a reasonably comprehensive article (or book chapter) explaining which basic theorems of ...
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303 views

Two-cardinal models of the random graph

For a first-order theory $T$ and cardinals $\kappa < \lambda$, we say that $M$ is a $(\kappa,\lambda)$-model if it is of size $\lambda$ and has a definable (with parameters) subset of size ...
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656 views

Semantic definition of sentence

This is a follow-up to question Completeness vs Compactness in logic 68788. One common theme was that compactness in logic is a purely semantic notion, so should have no need of completeness. The ...
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483 views

Intension vs. Extension: Coextensive relations in model and set theory

(originally posted at MSE as Same same but different: Coextensive relations in model and set theory, slightly modified) The official definition of a structure in model theory in its presumably most ...
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Completeness vs Compactness in logic

One standard approach to showing compactness of first-order logic is to show completeness, of which compactness is an easy corollary. I am told this approach is deprecated nowadays, as Compactness is ...
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220 views

a totally categorical structure with trivial geometry which is not interpretable in the trivial structure

Among the theorems of early geometric model theory there is one by Lachlan stating that every totally categorical structure with a trivial pregeometry is intrepretable in a dense linear order. That ...
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Which countable linear orders are $\aleph_0$-categorical?

The question is: Which countable linear orders are $\aleph_0$-categorical? I have a bit of progress on this: Define a discrete tuple to be a set of elements, ordered discretely, such that if $a$ and ...
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Homogeneous Structures

Let $M$ be a countable structure that is homogeneous, i.e. every isomorphism between finitely generated substructures of $M$ extends to an automorphism of $M$. Let $\phi_M$ be the Scott sentence of ...
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355 views

Higher-order preservation theorems?

The Łos-Tarski preservation theorem states that a set of formulas $F$ of first-order language $L$ is preserved under substructures for models of theory $T$ in $L$ precisely when $F$ is equivalent ...
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161 views

Classes which interpret any structure

I'm afraid this question might be too localized, but I have no better place to ask it: In the section Classes which interpret any structure of his Model Theory Hodges shows how each $L$-structure $B$ ...
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Definitions of definable compactness

We have an o-minimal structure M with the order topology. $X \subseteq M^n$ with the induced topology. The article "Definable compactness and definable subgroups of o-minimal groups" by Steinhorn and ...
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Why is definable compact equivalent to bounded and closed for sets with o-minimal structures?

We have $M$ an o-minimal structure. $X \in M^n$ with the induced topology. I'm reading an article which shows that $X \in M^n$ is definable compact is equivalent to $X$ being bounded and closed. ...
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391 views

A Dedekind (pseudo) finite set

Quoting the wiki:- a set A is Dedekind-infinite if some proper subset B of A is equinumerous to A. Explicitly, this means that there is a bijective function from A onto some proper subset B of A. -. ...
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Lindenbaum algebras and models

Sorry for this question out of the blue (especially if its answer should be trivial, obvious, or folklore): (When and how) can we construct models of a consistent first order theory $T$ from its ...
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Definition of a limit of a type

I apologize if this is not the right forum to ask such a basic question... In model theory what does that mean that a type concentrates on one point?
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Non-standard models of finite set theory

It is well known how the intended model and how the (countable) non-standard models of arithmetic look like. It's also well known how the intended model of set theory with the axiom of infinity ...
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an algebraically closed field definable in a real closed field

Is it true that an algebraically closed field $k$ definable (in the model theory sense) in a real closed field $\mathcal R$ is the algebraic closure $\mathcal{R}^{alg}=\mathcal{R}(\sqrt{-1})$? ...