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

learn more… | top users | synonyms

3
votes
1answer
296 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 ...
1
vote
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
votes
1answer
258 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
votes
1answer
163 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
votes
1answer
348 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
votes
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
331 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
votes
1answer
387 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
663 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
votes
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
votes
3answers
361 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 ...
-1
votes
1answer
297 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 ...
1
vote
1answer
209 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
votes
2answers
350 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
votes
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, ...
1
vote
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?
9
votes
2answers
692 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 ...
1
vote
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
votes
0answers
164 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
votes
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
votes
1answer
792 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
votes
2answers
415 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
vote
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
votes
0answers
77 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
votes
3answers
345 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 $M$ ...
8
votes
4answers
820 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
votes
2answers
451 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
227 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 ...
3
votes
2answers
347 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
votes
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
votes
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 ...
3
votes
0answers
151 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
votes
3answers
175 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
votes
1answer
706 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
286 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 ...
8
votes
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
444 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 ...
4
votes
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 ...
1
vote
0answers
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?
5
votes
2answers
475 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
votes
1answer
271 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
votes
1answer
216 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
votes
1answer
155 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 ...
5
votes
1answer
552 views

Countable admissible ordinals

Jensen claimed that for any finite increasing sequence countable admissible ordinals $\omega= \alpha_0<\alpha_1\cdots <\alpha_n$, there is a real $x$ so that, for each $m\leq n$, $\alpha_m$ is ...
10
votes
1answer
652 views

What is the etymology of model?

What is the etymology of model? The answer is of course pre-WWW, but the better part of an hour in the library searching both classic model theory and modal logic textbooks turned up nothing. Every ...
7
votes
0answers
348 views

Is the class of additive groups of rings axiomatizable?

I know that it is impossible to axiomatize the multiplicative structures of rings, called $R$-semigroups. Is anything known about the first-order axiomatizability of the class of abelian groups which ...
11
votes
1answer
372 views

Is ramification of number fields first order?

Fix a prime number $p$. Is there a first order sentence $\phi_p$ in the language of fields such that $\phi_p$ holds in a number field $K$ if and only if the prime $p$ is unramified in the field ...
5
votes
1answer
349 views

Elementary end extension of a countable model for ZF

Theorem 2.2.18 in Chang and Kiesler uses omitting types to show that any countable model of ZF has an elementary end extension. Can we control the countable order type of such a model? for example, if ...
7
votes
4answers
588 views

Reference Request: Non-Standard Models of PA

I am attempting to write an expository paper on non-standard models of PA that is accesible to students taking an introductory graduate course in mathematical logic (covering Godel's incompleteness ...
3
votes
2answers
521 views

Measure of progress towards a proof

Can one define some measure of progress towards a proof of a statement? I'm not sure if it's even possible for general first order logic statements so let's restrict ourselves to propositional ...