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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Using Lindstrom's theorem to prove Craig interpolation

[EDIT: The theorem I call "Beth definability" below is apparently not generally called that (wikipedia notwithstanding; see http://math.stackexchange.com/questions/288450/two-forms-of-beths-theorem). ...
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220 views

A canonical way to kill a subset of cohomology in a dg-algebra: via $A_\infty$-algebras? References?

Let $A$ be a differential graded algebra, $S\subset H^*(A)$. I would like to 'kill $S$ in a canonical way'. Is it possible to do it as follows: consider the $A_\infty$-algebra structure on ...
4
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1answer
167 views

When do substructures have computable copies?

Say that a class $\mathcal{C}$ of countable first-order structures in some finite signature has the effective substructure property if $\mathcal{C}$ is closed under isomorphism and whenever $A\in ...
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108 views

Compactness-like property for universal generalization?

Hi all! I have the following problem. Suppose I have a sequence of models $M_1,M_2,...$, all of which have the same countable domain (call it $D$). $x_1,x_2,...$ is a well-ordering of $D$. $\phi(x)$ ...
10
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2answers
442 views

Constructible models of New Foundations?

Hi all! Is there anything like Gödel's constructible universe for New Foundations? More precisely, I would like a process for taking a model $M$ of NF, and using it to build a model $L \subseteq M$ ...
4
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1answer
222 views

tennenbaum phenomena for the reals?

Let $\mathfrak{M} = \langle R, +,\times,> \rangle$ be such that $R$ is the set of real numbers and $\mathfrak{M} \models RA^1$ (the first-order axioms for the reals). Do we have characterisations ...
2
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216 views

Can we prove the completeness of FOL based on forcing?

I asked this question at http://math.stackexchange.com but I didn't get any answer there. In David Marker's Model Theory, there is a exercise said " we can view a countable Henkin construction as a ...
3
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1answer
300 views

Why Ryll-Nardzewski theorem fails for uncountable theories?

Ryll-Nardzewski theorems states that if $T$ is a countable complete theory, then $T$ is $\aleph_0$-categorical if and only if for every $n<\omega$ there are only finitely many formulas ...
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1answer
125 views

galois correspondence for hyperimaginaries

If there is a hyperimaginary $a_{E}$ and sets: $A,B\subset M$, such that for every automorphism F which fix A , F fix $a_{E}$, and for every automorphism F which fix B , F fix $a_{E}$. Does it true ...
6
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1answer
261 views

Non-definable elements vs indiscernible elements

Let $\Sigma$ be a one-sorted first-order signature, let $A$ be a $\Sigma$-structure, and let $B \subseteq A$ be a $\Sigma$-substructure. Fix a class $\mathcal{L}$ of formulae over $\Sigma$. We say an ...
4
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1answer
165 views

$L_{\omega_1,\omega}$ sentence with many automorphism in $\aleph_0$ and few automorphism in $\aleph_\omega$

Hello, I have the following question: Is it possible for a complete $L_{\omega_1,\omega}$ sentence $\phi$ to satisfy (a) the (unique) countable model of $\phi$ has $2^{\aleph_0}$ many automorphisms ...
2
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1answer
349 views

Decidability survives new constants

Let $L$ be a finite first order language and let $M$ be an $L$-structure with universe $\mathbb{N}$ that interprets all $L$-symbols as recursive sets (so $M$ is a recursive $L$-structure). Let ...
6
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1answer
435 views

(Finite) Models of two subtheories of Peano Arithmetic

Consider first-order theory (with identity) of Peano Artithmetic built in the language $\{S,+,\times,0\}$ and with the following set of axioms: \begin{align} \neg Sx&=0\tag{1}\\\ ...
5
votes
1answer
332 views

Smallest real closed field realizing all cuts of the rational numbers

Let $K$ be a real closed field of transcendence degree 1 over $\mathbb{R}$. It is not difficult to see that $K$ has the following "minimality property": Whenever $L$ is a real closed field that ...
10
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1answer
396 views

Number of linear orders

It is well known that for every infinite cardinal $\kappa$ the number of non-isomorphic total orders of cardinality $\kappa$ is $2^\kappa$. Who first proved this, and in what context? Was it proved ...
4
votes
3answers
682 views

Survey of finite axiomatizability for relational theories?

An $L$-theory $T$ is finitely axiomatizable if there is a finite set $A$ of $L$-sentences with the same consequences as $T$, i.e. such that $M \models T$ iff $M \models A$ for every $L$-structure $M$. ...
3
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1answer
203 views

Nondegenerate involutions on the complex numbers — always just conjugation?

Suppose you have a ring homomorphism $(-)':\mathbb{C} \to \mathbb{C}$, which is an involution such that $\sum_i a_i a_i' = 0 \Leftrightarrow \forall i \ a_i=0$, where this sum is finite. Must this ...
2
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3answers
365 views

Methods for proving non FO definability

I'm having difficulties to prove that the subset of even numbers is not first-order definable in $(\mathbb{N},<)$. Any hint is welcomed! More generally, what are usual techniques in order to prove ...
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1answer
303 views

A question on intuitionistic propositional logic [closed]

One week ago, I asked a question on math.stackexchange.com (http://math.stackexchange.com/questions/209120/a-question-on-intuitionistc-propositional-logic). But nobody answered my question. So I ...
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1answer
215 views

Existentially closed substructure and ultraproducts

Is it true that if M is existentially closed in N then N can be embedded in an ultraproduct of M ? Best regards.
2
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2answers
354 views

Applications of the Ax Kochen Ershov (AKE) princicple

The AKE priciple states that two finitely ramified Henselian field (this means that the field is either of residual caracteristic 0 or is of characteristic $p$ and there is only a finite number of ...
0
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1answer
155 views

Nonsingular zeroes are algebraic?

I'm getting started in Real Algebraic Geometry (from a model-theory perspective), and a paper makes the following assertion (here $K\subset L$ are real-closed fields): Suppose $Q\in L^n$, $f_1, ...
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2answers
321 views

How do we avoid circularity when we build a structure for ZFC? [closed]

when investigating ZFC as a formal language a structure is a set, are we not engaging in circular logic here? Or is 'set' thought of in a more primitive sense?
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2answers
703 views

Intuition behind o-minimal structures.

This is very much the same post as I posted at math.stackexchange. I am following the definitions of an o-minimal system in "Tame Topology and O-minimal Structures" by Lou van den Dries. It is ...
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0answers
169 views

Characterizing $\mathbb{Q}$ among number fields

Is there an $\mathcal{L}_{\omega_1\omega}$ formula or set of formulas that characterizes the rationals $\mathbb{Q}$ among other number fields? EDIT: My formula must not contain an infinite number of ...
2
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0answers
168 views

Is this fragment of arithmetic on $p^{-\infty} {\mathbb Z}$ decidable?

Let $p$ be a prime number. Consider the abelian group $p^{-\infty} {\mathbb Z} = \bigcup p^{-n} {\mathbb Z}$ consisting of rational numbers whose denominator is a power of $p$, under addition. View ...
3
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1answer
356 views

Interpretations as morphisms

Model-theoretic interpretations seem to give rise to categories in which morphisms are not functions. To name just two small examples: the category of finite graphs with interpretations between them ...
6
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1answer
793 views

Where do nonstandard elliptic curve angles come from?

This is a question which has bounced around my head over the past few years. At the same time, I am answering ...
5
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2answers
421 views

generalizing the ultrapower

Given an 2-valued measure $\mu$ on a set $I$, and structures $M_i \ (i \in I)$, one can construct the ultrapower $\prod_{i\in I}M_i / U$ (where $U$ is the ultrafilter associated with $\mu$.) One can ...
1
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1answer
333 views

Riemann hypothesis for zeta function of definable sets over finite fields

Hi, Consider the zeta function of a definable set over a finite field. More precisely, let $\varphi$ be a formula in the language of fields and let $X$ be a definable subset of $F^n$ given by ...
4
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78 views

local character of Tarski-Vaught for tuples in excellent classes

In the book Baldwin, Categoricity in Abstract Elementary Classes defines (Def.20.1,p.151) a notion of Tarski-Vaught extensions for tuples that generalises both independence and usual Tarski-Vaught ...
6
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3answers
349 views

computing Morley rank using parameters from an arbitrary model

One of the ways to define the Morley rank of a definable set is with respect to a model, say $M$, i.e. a set has rank $\alpha+1$ if there are infinitely many definable subsets with parameters in $M$ ...
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4answers
825 views

Axioms for Riemann $\zeta$ function

Are there any set of axioms that completely characterize the Riemann zeta function? i.e. like Ressayre axioms for the exponential function in an exponential field or functional equations.
3
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2answers
459 views

Cantor theorem on orders

It is "a well-known theorem of Cantor", said Sierpinski (circa 1920), that every countable total order can be imbedded in the rationals, and he proceeds to demonstrate that, assuming the continuum ...
4
votes
2answers
229 views

an example of a strictly superstable field

It is well-known that the theory of separably closed fields of some fixed positive characteristic and degree of imperfection is stable but not superstable. By a result of Cherlin and Shelah a ...
2
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2answers
352 views

The category of Boolean-valued models associated to a model of ZFC

This is related to my previous question here, and indirectly motivated by Andreas Blass intriguing answer therein: start from a ZF transitive model $M$, and consider the category $CBA(M)$ of ...
2
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2answers
613 views

All properties of a mathematical object

This is primarily a question about related literature. I am looking for specific references, or terminology that I can use to search for references. Let A a well defined mathematical structure of ...
2
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1answer
143 views

Feferman-Kreisel preservation theorem

I want to show the following theorem from Feferman and Kreisel: Let $\phi$ be such that there is a theorem $\sigma$ of ZFC so that for every transitive model $M$ of $\sigma$, we have: $\phi^M \to ...
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152 views

axiomatizing the abelian part of the topological fundamental groupoid functor on algebraic varieties

let Vv be the category of complex algebraic varieties defined over $\bar Q\subset C$, and let $\pi_1^{top}:Vv\longrightarrow Groupoids$ sending a variety V into its (strict) fundamental groupoid ...
2
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3answers
176 views

Is there a general theory of models that has as instances classical FOL, classical propositional logic, etc.?

Is there any general theory of models that has as instances classical FOL, classical propositional logic, etc.?
11
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1answer
719 views

Elementary Equivalence =? Homotopy Equivalence

One of the most interesting novelties in recent foundational studies is Voevodsky's Homotopical Type Theory project (see here). Finally homotopy theory ideas have entered in a royal fashion the ...
4
votes
3answers
287 views

Can every $\mathcal{L}_{\omega_1,\omega}$ formula be expressed as a type? What about canonical forms?

If $\mathcal{L}$ is a countable, first-order language, it is easy to see that every $n$-type $p$ (over $\emptyset$) can be expressed as an $\mathcal{L}_{\omega_1,\omega}$-formula, namely ...
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0answers
243 views

Uncountable Lüroth problem

Question. Let $F(X)$ be the field obtained by adding an uncountable collection of indeterminates (mutually transcendental elements) to a prime field $F$. Is there an example of a subfield $E$ ...
11
votes
1answer
447 views

How badly does compactness fail in $\mathcal{L}_{\omega_1\omega}$?

I would like to get a better idea of how badly compactness fails in $\mathcal{L}_{\omega_1\omega}$. Let $\Gamma$ be an arbitrary set of sentences from $\mathcal{L}_{\omega_1\omega}$. Let the ...
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1answer
346 views

Is there a statement equivalent to a sentence admitting $(\alpha^{+n},\alpha)$?

From Chang and Keisler's "Model Theory", section 7.2, we know that: 1) There is a sentence $\sigma$ in a suitable language $L$ such that for all infinite cardinals $\alpha$, $\sigma$ admits ...
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201 views

Open question in abstract model theory

The existence of an extension of first order logic satisfying both the Compactness Theorem and the Interpolation Theorem is an open or solved question?
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2answers
487 views

Are there any complete, first-order and unstable theories which have non-categorical second-order formulations?

Since it's not stable, $PA$ fails at being categorical in a power in the worst possible way, having $2^{\lambda}$ models in any uncountable $\lambda$. But $PA$ regains its categoricity in the move to ...
6
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1answer
274 views

Is there exponentiation in “sufficiently large” models of $I\Delta_{0}$?

Let $L_{E}$ be the language of discretely ordered rings together with an extra predicate symbol $E$. The system $A$ consists of the axioms of $I\Delta_{0}$ (basic arithmetic plus induction for bounded ...
0
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1answer
218 views

Non-Archimedean non-standard models for R

Let $\langle \mathbb{R}, 0, 1, +, \cdot, <\rangle$ be the standard model for $R$, and let $S$ be a countable model of $R$ (satisfying all true first-order statements in $R$). Is it true that the ...
2
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1answer
157 views

Terminology for system of equations and…

I am looking for the standard term for a system that consists of things of the form $p_i(x_1,\ldots ,x_n)=0$ and of the form $q_j(x_1,\ldots,x_n)\neq 0$ with the $p_i$ and $q_j$ polynomials. I have ...