**26**

votes

**0**answers

588 views

### Where do uncountable models collapse to?

Suppose $T$ is a complete first-order theory (in an finite, or at worst countable, language). Given any model $\mathcal{M}\models T$ of cardinality $\kappa$, we can ask whether $\mathcal{M}$ can be ...

**17**

votes

**0**answers

850 views

### Hrushovski's Construction

Zilber expressed a conjecture for $\aleph_{1}$- categorical theories (In the 80s).
Zilber's Conjecture: The geometry of any $\aleph_{1}$- categorical structure is one of the following:
(a) Trivial (...

**15**

votes

**0**answers

924 views

### Defining $\mathbb{Z}$ in $\mathbb{Q}$

It was proved by Poonen that $\mathbb{Z}$ is definable in $\mathbb{Q}$ using $\forall \exists$ formula. Koenigsmann has shown that $\mathbb{Z}$ is in fact definable by universal formula. What is the ...

**14**

votes

**0**answers

374 views

### What is the Cantor-Bendixson rank of the space of first order theories?

Let $L$ be the language $\{R\}$ containing a single binary relation symbol. Consider the space $S_0(L)$ of complete, first-order $L$-theories. This is a seperable, compact Hausdorff space; what is its ...

**10**

votes

**0**answers

187 views

### Example of $\aleph_1$-categorical linear order

Is it possible to have an $L_{\omega_1,\omega}$-sentence $\phi$ in a vocabulary that includes $<$ that satisfies the following?
$<$ is a linear order on a definable subset;
$\phi$ is $\aleph_1$...

**10**

votes

**0**answers

295 views

### A question concerning model theory of groups

Several days ago, Professor Martin Bridson gave a very nice talk in my department. Some questions concerning his talk came into my brain Since I am neither a model theorist nor a algebraist, I am not ...

**10**

votes

**0**answers

205 views

### Maximality of linear orders in the Keisler order on theories

Recently Malliaris and Shelah (see their preprint http://math.uchicago.edu/~mem/Malliaris-Shelah-CST.pdf) have shown that theories with $SOP_2$ are maximal in the Keisler order. A preceding result of ...

**10**

votes

**0**answers

233 views

### Stability of analytic Zariski structures

Noetherian Zariski structures are introduced by Hrushovski and Zilber(1996).
An analytic Zariski structure is a generalization of Noetherian Zariski structures, introduced by Zilber and Peatfield.
...

**10**

votes

**0**answers

212 views

### Krull dimension and Morley rank

Definition : A Topological space $\mathcal{D}$ is called noetherian if it satisfies the descending chain condition for closed subsets. We define the dimension of $\mathcal{D}$ to be the supremum of ...

**9**

votes

**0**answers

315 views

### Elementary proof of Chevalley's Theorem on constructible sets

I am looking for a proof of the easiest affine version of Chevalley's Theorem on constructible sets :
Theorem (Chevalley). The image of a constructible subset of $\mathbf C^n$ by a polynomial map $P:...

**9**

votes

**0**answers

364 views

### Riemann hypothesis in Zilber's field

Question. What is known about the situation (truth or falsity) of Riemann hypothesis in the Zilber's field?

**9**

votes

**0**answers

159 views

### From interpolation to separation

Lusin's separation theorem states that, if $A$ and $B$ are disjoint analytic subsets of a Polish space, then there is a Borel set $X$ separating them ($A\subseteq X$, $B\cap X=\emptyset$). Craig's ...

**9**

votes

**0**answers

291 views

### “Fraïssé limits” without amalgamation

All structures are countable with countable signature.
Given a structure $\mathcal{A}$, the age of $\mathcal{A}$, $Age(\mathcal{A})$, is the set of structures isomorphic to finitely-generated ...

**9**

votes

**0**answers

348 views

### 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). ...

**8**

votes

**0**answers

191 views

### ω-categorical, ω-stable structure with trivial geometry not definable in the pure set

Briefly, my question is the following.
does every countable ω-categorical, ω-stable structure with
disintegrated strongly minimal sets interpret in the countable pure set?
By countable pure set I ...

**8**

votes

**0**answers

108 views

### Two cardinal obstructions

Given a theory $T$ and a formula $\phi(x)$ we say that they admit a $(\kappa, \lambda)$ model if there is a model $M$ such that $|M| = \kappa$ and $|\phi(M)| = \lambda$.
In all examples that I know ...

**8**

votes

**0**answers

221 views

### Is there a notion analogous to separability but requiring definable rather than countable sets?

Among models of $\lambda$-calculus, some like the Bohm tree model have the property that every element is a directed sup of definable elements, whereas others like the $D_\infty$ and $P(\omega)$ ...

**8**

votes

**0**answers

250 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 $E$ of $F(X)$...

**7**

votes

**0**answers

130 views

### Kripke models of $HA$

Let $K$ be a kripke model and $k$ be one of its node, then $\mathcal{M}_k$ is classical structure of $k$.
What is the strongest theory of arithmetic like $T$ such that for
every kripke model $K\...

**7**

votes

**0**answers

101 views

### Is there a ``Ladner's Theorem" for the PH-vs-PSPACE scenario?

Like a statement of the kind, ``If the Polynomial Hierarchy (PH) $\neq$ PSPACE then there exists $L \in PSPACE \backslash PH$ which is not PSPACE-complete"?
Or is there something else that states ...

**7**

votes

**0**answers

107 views

### What classes of complex manifolds are known to be definable in an o-minimal expansion of the real field?

It is a widely known (perhaps slightly folkloric) fact that compact complex manifolds, understood as first-order structures with a predicate for each analytic subset, are definable in an expansion of ...

**7**

votes

**0**answers

210 views

### Is there a Rado category?

The Rado graph appears to have a nice universality property (it contains all finite and all countably infinite graphs as induced subgraphs) and homogeinety property (any isomorphism between finite/...

**7**

votes

**0**answers

2k views

### 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.
...

**6**

votes

**0**answers

100 views

### Finitely presented algebras with isomorphic semilattices of congruences

Let $\mathbb{T}$ be a finitary algebraic theory. For each $\mathbb{T}$-algebra $A$, let $Q (A)$ be the join semilattice of finitely generated congruences on $A$. There is an evident pushforward ...

**6**

votes

**0**answers

122 views

### model theory of non-reduced schemes

In model theory one studies Boolean algebras of definable sets of complete theories. For many theories definable sets are in direct correspondence with geometric objects, for example, definable sets ...

**6**

votes

**0**answers

100 views

### Can the isomorphism relation for countable models become harder when adding finitely many constants?

I am particularly interesting in the case where $T$ is o-minimal, but I would be interested in any theory $T$ (or even an $L_{\omega_1,\omega}$-sentence) which has this property.
Context: view the ...

**6**

votes

**0**answers

269 views

### Is there Ultracoproduct-like construction for topological spaces in general?

In
http://arxiv.org/pdf/math/9704205.pdf
they define the ultracoproduct of a sequence of compact Hausdorff spaces, $\sum_\mathcal{U}X_i$ along an ultrafilter $\mathcal{U}$ as the Wallman-Frink ...

**5**

votes

**0**answers

86 views

### Does every model of $I\Delta_0$ has an end extension to a model of $I\Delta_0+\Omega_1$?

Does every model of $I\Delta_0$ has an end extension to a model of $I\Delta_0+\Omega_1$?
In End extensions of models of linearly bounded arithmetic paper the author said this problem is open. I want ...

**5**

votes

**0**answers

107 views

### Lascar strong types in fragments of arithmetic

Are Lascar strong types (definition below) in models of fragments of arithmetic always type definable? (They trivially are, in models of full induction.)
Definition Given a saturated model ${\cal M}$ ...

**5**

votes

**0**answers

109 views

### O-Minimal sentences in $L_{\omega_1,\omega}$?

Is there any meaningful sense in which we can talk about o-minimal sentences of $L_{\omega_1,\omega}$? I can give a first attempt, easily; given a countable fragment $F$ and a sentence $\Phi$ in that ...

**5**

votes

**0**answers

206 views

### Universal anti-Horn classes?

Is there published work about universal anti-Horn classes?
Anti-Horn formulas are also sometimes known as dual Horn.
See also related question Is there any research of universal algebras axiomatized ...

**5**

votes

**0**answers

385 views

### Recent application of model theory in set theory by Shelah-Malliaris

Recently one of the oldest open problems in set theory about the cardinal invariants of the continuum (i.e the question of whether $\mathfrak{p}=\mathfrak{t}$) was solved by Shelah and Malliaris (see "...

**5**

votes

**0**answers

207 views

### Schanuel's conjecture and abstract elementary classes

Are there any connections between Schanuel's conjecture and abstract elementary classes. More precisely
Question. Is there any conjecture in abstract elementary classes whose truth implies the ...

**5**

votes

**0**answers

912 views

### Undecidability degree of some elementary theories (two equivalence relations, …)

I have a question about some results in the paper
I. A. Lavrov. Effective inseparability of the sets of identically true formulae and finitely refutable formulae for certain theories (in Russian). ...

**4**

votes

**0**answers

168 views

### Hyperimaginaries and continuous logic

Classical (i.e., discrete) logic is well positioned to study imaginaries in part because the $T^{eq}$ construction allows us to treat imaginary sorts as we would treat any other sort. With ...

**4**

votes

**0**answers

128 views

### Is the lowenheim-skolem number of nth order logic larger than the corresponding number for 2nd order logic

According to this paper, by Vaananen, the $LS$ number for $2^{nd}$ order logic is given by "the supremum of $Π_{2}$-definable ordinals", where "The Lowenheim-Skolem number $LS(L)$ of $L$ is the ...

**4**

votes

**0**answers

112 views

### Fixed Points of the Friedman Stanley Jump

Consider the situation of a pair $(X,E)$, where $X$ is a standard Borel space and $E$ is an invariant equivalence relation on $X$*. The Friedman-Stanley jump of this pair is an equivalence relation $...

**4**

votes

**0**answers

180 views

### Relation between fundamental theorem of Ehrenfeucht-Fraisse games and notions of bisimulation and simulation

Vijay D alluded to the relation between the fundamental theorem of Ehrenfeucht-Fraisse games and notions of bisimulation and simulation in response to the whats-a-magical-theorem-in-logic question.
...

**4**

votes

**0**answers

105 views

### Stabilization of recursive approximation in $PA^-+I\Sigma_1^0$

Over any model M of $PA^-+I\Sigma_1^0$. Suppose $A\in [T]$ where $T$ is a $\Delta_2^0$-tree and $A$ is one isolated path. Further, $A$ is regular, i.e. $\forall n A\upharpoonright n$ has a code in $M$....

**4**

votes

**0**answers

191 views

### strong order property in continuous logic

I am wondering what is the accepted version of the strong order property in continuous logic.
The definition for classical logic is as follows:
$T$ has SOP$_n$ (for $n\geq 3$) if there is a formula $\...

**4**

votes

**0**answers

199 views

### Saturated Ehrenfeucht-Mostowski models

Inspired by this question on MSE I tried to prove the following:
Let $T$ be a complete theory in a first order language and $\kappa$ a cardinal. If $T$ is $\kappa$-stable, then there exists a $\...

**4**

votes

**0**answers

222 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 $H^\ast(A)$,...

**4**

votes

**0**answers

80 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 ...

**4**

votes

**0**answers

177 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
$s,t:...

**4**

votes

**0**answers

431 views

### 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 $M$...

**4**

votes

**0**answers

312 views

### Sentences Preserved by Direct Products (including the Empty Product)

Consider the class of all structures for a given signature in first-order logic. Let $S_i$ be a family of structures, and $\oplus S_i$ be the direct product of the family. You can extend the notion ...

**4**

votes

**0**answers

463 views

### Natural models of graphs?

Motivation
I want to capture the notion of natural models of finite graphs: How can natural predicates and natural relations on a given natural base class $D$ be defined? If this succeeds the ...

**3**

votes

**0**answers

164 views

### Construction of model of arithmetic from an arbitrary model

Let $M$ be a non-standard model of $PA$, $a\in |M|$ be an arbitrary non-standard number and $T$ be a theory of arithmetic. We want to choose a subset $M'\subsetneq M$ such that:
$M'\models PA^-$ (or ...

**3**

votes

**0**answers

108 views

### Limits of definable maps

For sequences of semialgebraic maps there is the following result:
Let $(f_{n}: ]0,1[^d \to ]0,1[)_{n \in \mathbb{N}}$ be a sequence of continuous semialgebraic maps of bounded degree such that $(...

**3**

votes

**0**answers

94 views

### Does Łoś's theorem imply choice given a free ultrafilter?

In the paper "Łoś's theorem and the boolean prime ideal theorem imply axiom of choice" Howard has shown that Łoś's theorem and the boolean prime ideal theorem imply axiom of choice. At the end of the ...