**14**

votes

**0**answers

815 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 ...

**12**

votes

**0**answers

742 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 ...

**10**

votes

**0**answers

166 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 ...

**8**

votes

**0**answers

212 views

### Main Gap Phenomenon

Shelah's Main Gap Theorem states that for all first-order, complete theories, T, in a countable language, we have that either $$I(T,\aleph_\alpha)=2^{\aleph_\alpha}$$ or ...

**8**

votes

**0**answers

301 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). ...

**7**

votes

**0**answers

193 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 ...

**7**

votes

**0**answers

190 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 ...

**7**

votes

**0**answers

236 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 $E$ ...

**7**

votes

**0**answers

334 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 ...

**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

185 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 $P(\omega)$ ...

**6**

votes

**0**answers

175 views

### Foundation scheme for $\Sigma_{n+1}$-formulas

I have trouble working out a proof in the second part of
Jean-Pierre Ressayre and Alex Wilkie. Modèles non standard en arithmétique et théorie des ensembles. Publications ...

**6**

votes

**0**answers

91 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 ...

**6**

votes

**0**answers

177 views

### Axiomatizations of the real exponential field

According to Marker's "Model Theory: An Introduction", the real exponential field has a $\forall\exists$ axiomatization (because it is model complete) but no-one has any idea what such an ...

**6**

votes

**0**answers

233 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

327 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

174 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 ...

**5**

votes

**0**answers

87 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 ...

**5**

votes

**0**answers

890 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). ...

**5**

votes

**0**answers

428 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 ...

**4**

votes

**0**answers

112 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

88 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 ...

**4**

votes

**0**answers

208 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.
...

**4**

votes

**0**answers

96 views

### Cardinality of Grothendieck ring in model theory

Good evening,
In model theory there is a notion of Grothendieck ring defined here http://math.berkeley.edu/~scanlon/papers/greu12jun00.pdf.
Do we know anything about the cardinality of these rings ?
...

**4**

votes

**0**answers

193 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 ...

**4**

votes

**0**answers

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**

votes

**0**answers

75 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

299 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 ...

**3**

votes

**0**answers

147 views

### Equational theories determined by “identities without variables”

How to characterize equational theories $T$ which have the following property: for any two terms $t(x_1,...,x_n)$ and $t'(x_1,...,x_n)$ in the signature of $T$, if for any closed terms (i. e. terms ...

**3**

votes

**0**answers

119 views

### Peano (Dedekind) categoricity

What is the smallest fragment of second order logic such that $Th(\mathbb{N})$ in that logic is categorical (only one model, namely natural numbers, up to isomorphism). For example, can we do this in ...

**3**

votes

**0**answers

85 views

### Is a proconstructible subsemigroup of $M_n(\mathbb{C})$ an intersection of constructible subsemigroups?

Let $S$ be a proconstructible subsemigroup of $M_n(\mathbb{C})$, that is a subsemigroup which is an intersection of constructible sets. Is $S$ an intersection of constructible subsemigroups?
The ...

**3**

votes

**0**answers

164 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 ...

**3**

votes

**0**answers

158 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 ...

**3**

votes

**0**answers

162 views

### Is there a useful Galois connection between Languages and Grammars?

I've just beginning to learn logic and proof theory - and the following rather vague and perhaps ill-formed question occurred to me.
Given an alphabet it's straightforward to construct the Language, ...

**3**

votes

**0**answers

149 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
...

**3**

votes

**0**answers

300 views

### On Grothendieck ring and semiring

We are given a language $L$ and a structure $M$ (model). Definable sets in this model are subsets of $M^n$ definable by a formula of $L$.
The Grothendieck semiring of the structure is defined in the ...

**3**

votes

**0**answers

158 views

### 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 ...

**3**

votes

**0**answers

372 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 ...

**3**

votes

**0**answers

402 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 ...

**3**

votes

**0**answers

276 views

### To what extent MSO = WS1S, when adding relations?

Let me first clarify my definitions. For a word $w \in \Sigma^*$, with $\Sigma=\{a_1, \ldots, a_n\}$, I define two structures:
$${\mathbb{N}}(w) = \langle {\mathbb{N}}, <, Q_{a_1}, \ldots, Q_{a_n} ...

**2**

votes

**0**answers

85 views

### motivic integration and jacobian ideal

When we consider the change of variables in motivic integration, we have a birational map $f:Y\rightarrow X$ with Y smooth and we have to consider two invariants the order of the Jacobian ideal of $X$ ...

**2**

votes

**0**answers

52 views

### Characterization of externally definable sets

Let $\cal U$ be a saturated model of inaccessible cardinality $\kappa$. For arbitrary $\cal D\subseteq U$ denote by $\langle\cal U,D\rangle$ the expansion of $\cal U$ with a new predicate for $\cal ...

**2**

votes

**0**answers

104 views

### Is the following a sufficient condition for being a primal algebra?

I have a question regarding universal algebra and, in particular, primal algebras:
Suppose that A is a finite simple algebra with no proper subalgebra, no automorphism except the identity map, with a ...

**2**

votes

**0**answers

117 views

### Preimages of accessible full subcategories

My question is ultimately about the model theory of $L_{\infty, \omega}$, but it is more convenient to phrase it in terms of category theory. Suppose I have finitely accessible categories ...

**2**

votes

**0**answers

42 views

### quantifier rank and number of variables as complexity measures

Is there a property of finite structures expressible with a sentence using only 2 variables and quantifier rank n but not expressible by any sentence with more variables and quantifier rank less than ...

**2**

votes

**0**answers

152 views

### Algebras admitting quantifier elimination

I apologize if this question is meaningless or trivial:
What are examples of Algebras admitting quantifier elimination? Especially are there Groups admitting quantifier elimination?
I need to say ...

**2**

votes

**0**answers

366 views

### Deducing Skolem's nonstandard integers from downward Lowenheim-Skolem?

If one has a nonstandard model $\mathcal{N}$ of PA and adjoins to the first-order theory the countable list of axioms $1<H,\, 2<H,\, 3<H, \ldots$ (satisfied in $\mathcal{N}$) for all the ...

**2**

votes

**0**answers

124 views

### topology generated by irreducible componets of $\Gamma$-invariant closed sets

For an analytic space $U$ equipped with an action of a group $\Gamma$,
call a subset $Z\subseteq U$ $\Gamma$-closed iff
it is a closed analytic subset and each of its irreducible components
is an ...

**2**

votes

**0**answers

214 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 ...

**2**

votes

**0**answers

161 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 ...