**6**

votes

**1**answer

118 views

### Is nfcp equivalent to stable + eliminates $\exists^\infty$?

Let $T$ be a complete first-order theory. Recall that a formula $\phi(\overline{x},\overline{y})$ has the finite cover property (fcp) if for all $n$, there exist $\overline{a}_1,\dots,\overline{a}_n$ ...

**12**

votes

**1**answer

518 views

### Is there a nonstandard model of arithmetic having precisely one inductive truth predicate?

$\newcommand\Tr{\text{Tr}}$My question is whether there can be a nonstandard model of PA having a unique inductive truth predicate.
Background. If $\mathcal{N}=\langle N,+,\cdot,0,1,<\rangle$ is ...

**3**

votes

**1**answer

330 views

### Are there sets which are computable in one model, but uncomputable in another?

Suppose we have two models of set theory, $U$ and $V$ which have the same $\Bbb N$. Is it possible that there is a set $A\subseteq\Bbb N$ such that, in $U$, this set is computable, i.e. there is a ...

**6**

votes

**1**answer

203 views

### O-minimal Theories with Non-Dense Order Type

I asked this question on MSE, but I haven't received any comments or responses (also, it has a very low view count), so I thought I would also ask it here.
In this paper, Knight, Pillay, and ...

**10**

votes

**2**answers

465 views

### Is every order type of a PA model the \omega of some ZFC model?

Let $N$ be a model of first-order Peano arithmetic, and let $\sigma$ be its order-type. Does it follow that there is a (non-transitive, expect when $M$ is the standard model) $ZFC$-model $M$ such that ...

**3**

votes

**0**answers

100 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 $X$ ...

**1**

vote

**1**answer

113 views

### Saturated models and definable substructures

Let $M$ be a saturated model of a theory $T$ in a first-order language $\mathcal{L}$, and let $N$ be a submodel of $M$.
Is it possible to have a substructure $A\neq N$ of $M$ such that $N \subset A ...

**1**

vote

**1**answer

174 views

### Two questions about axiomatic rank of groups

Let $G$ be a group and $V=Var(G)$ be the variety generated by $G$. Suppose the axiomatic rank of $V$ is $n$. Let $Id(V)$ be the set of all identities of $V$.
1- Can we say that every element of ...

**1**

vote

**0**answers

90 views

### Real algebraic groups and pseudo-finiteness

What is the relationship between pseudo-finite groups and real algebraic groups?
Could you provide an example of a pseudo-finite real algebraic group and of a non pseudo-finite one, if any?
Thank ...

**13**

votes

**1**answer

372 views

### Groups and pregeometries

Definition.
For an infinite structure $\mathcal{A}$ and $cl : P(dom(\mathcal{A})) \longrightarrow P(dom(\mathcal{A}))$ , we say
that $(\mathcal{A}, cl)$ is a structure carrying an $\omega$-homogeneous ...

**2**

votes

**0**answers

68 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 ...

**6**

votes

**3**answers

304 views

### Does this property of a first-order structure imply categoricity?

Let $\mathfrak{A}$ be a first-order structure over a relational language and let $\kappa$ be an infinite cardinal. Lets say that $\mathfrak{A}$ has the $\kappa$-property if for every structure ...

**4**

votes

**1**answer

181 views

### Are there superexponential Pfaffian functions?

This question is motivated by model theory, but it's really an analysis question (which means it may have an easy analysis answer that I just don't have the background for). Here's the main question, ...

**13**

votes

**3**answers

684 views

### Can Suslin (or Aronszajn) lines ever be orderings of abelian groups?

I am interested in realizing linear orders as orderings of abelian groups. In particular, can Suslin lines (and other classes of line) be realised in this way?
Let $\mathcal{C}$ be a class of ...

**1**

vote

**1**answer

100 views

### dual (p,q)-property

If the set system $(X,S)$ has the $(p,q)$-property does its dual system also have the property? (Possibly, for different $p$ and $q$.)
Explicitly, I am asking about the equivalence of the following ...

**8**

votes

**1**answer

297 views

### Saturated Ultrapowers

I posted it on MSE and didn't receive any comments or answers, so I thought I would post it here.
(See http://math.stackexchange.com/questions/895549/keisler-order-saturated-ultrapowers)
Keisler's ...

**2**

votes

**1**answer

167 views

### References for the Keisler Order

Are there any good/modern references on the Keisler order. I have been reading Keisler's original paper, "Ultraproducts which are not Saturated", which introduces the order. However it is somewhat ...

**1**

vote

**0**answers

77 views

### definable relations in finite structures

Let $\mathcal{L}$ be a first order language and $M$ be a finite $\mathcal{L}$-structure. I want to know which relations $R\subseteq M^n$ are definable in $\mathcal{L}$? If I apply the well-known ...

**3**

votes

**1**answer

98 views

### Is quasivariety generated by all perfect graphs finitely axiomatizable?

Fix logic $L$ with equality and a binary relation symbol $E$.
The class of graphs can be identified with the class of models of the universal first-order Horn $L$-sentences $\forall x,y\; E(x,y) ...

**4**

votes

**1**answer

197 views

### Countable model theory for $\omega$-stable theories?

This is a bit of a fishing expedition, because I'm not sure what I'm looking for. Very vaguely stated, here's the driving question:
What conditions on an $\omega$-stable theory make the class of ...

**5**

votes

**1**answer

317 views

### How do we know if Vaught's Conjecture is Absolute?

Please note that this might be some confusion on my part about the work surrounding Vaught's conjecture.
First of all, Vaught Conjecture states that if a first-order complete theory $T$ in a ...

**12**

votes

**1**answer

409 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 ...

**3**

votes

**2**answers

559 views

### A double centralizing theorem for finite groups

I have a proof for the following assertion which employs Model Theory. It has certainly a pure group theoretic proof, but what is such a proof? Is the assertion trivial?
Theorem Let $G$ be a finite ...

**4**

votes

**1**answer

203 views

### Elementary chains of $\aleph_1$-saturated models

If $X$ and $Y$ are two sets linearily ordered by $<$, $X$ is called cofinal in $Y$ if $X \subseteq Y$ and and for every $y \in Y$, there is a $x \in X$ with $y < x$.
If $M$ is some model and ...

**5**

votes

**1**answer

275 views

### What axioms (other than choice) have a taming effect on the ordering of cardinalities?

Axiom of choice arranges all cardinalities into a well-ordered chain but without it their ordering can be wild in general ZF models, e.g. two cardinalities may not even have inf or sup. However, ...

**8**

votes

**1**answer

187 views

### The number of countable models [duplicate]

Let $\mathcal{L}$ be a countable first order language. For a natural number n, can we find a complete $\mathcal{L}$-Theory $T$ which has exactly n non-isomorphic countable models ?

**4**

votes

**1**answer

229 views

### Isomorphism of real closed fields

Given two real closed fields $R_1$ and $R_2$ such that both have cardinality continuum, archimedean, but not necessarily complete. Assume further that they are back and forth equivalent (in the ...

**7**

votes

**2**answers

424 views

### A “suitably generic” set of Cohen reals without forcing?

I was reading a paper by David Marker, whose main theorem was that if $T$ is a first-order theory which is not small, then $F_2\leq_B \cong_T$. That's not especially relevant to the question at hand, ...

**6**

votes

**2**answers

201 views

### Models of PRA/EFA with induction on $X$ but not $\omega^X$

As I currently understand it, induction on formulas containing $N+1$ first-order quantifiers is required to prove the well-ordering of the ordinal $(\omega \uparrow\uparrow N) < \epsilon_0$, that ...

**4**

votes

**1**answer

143 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 ...

**4**

votes

**1**answer

92 views

### Counterexample for closedness under union of $\prec_{\infty,\kappa}$ chains

Assume $\kappa$ is uncountable and $\phi$ is an $L_{\infty,\kappa}$ sentence. Let $K$ be the collection of models of $\phi$ partially ordered by $\prec_{\infty,\kappa}$. It is well-known that $K$ is ...

**4**

votes

**3**answers

149 views

### A model with $\kappa$ many automorphism and a rigid element.

The following should be known, but I could not find an example.
Let $\kappa$ be an uncountable cardinal. Find a model $M$ of size $\kappa$ which has $\ge\kappa$ many automorphisms, but for some $m\in ...

**1**

vote

**1**answer

146 views

### How can one define the direct limit of classes?

If we have a family of classes $(\mathfrak{M}_\alpha)_{\alpha\in D}$ of $\in$-structures with $D$ being a limit ordinal or the class of ordinals, and a family ...

**24**

votes

**3**answers

2k views

### Interactions between (set theory, model theory) and (algebraic geometry, algebraic number theory ,…)

Set theory and model theory have many applications outside of logic, in particular in algebra, topology, analysis, ...
On the other hand model theory, in particular after Hrushovski, found many ...

**4**

votes

**2**answers

363 views

### Overspill in models of arithmetic

Assume that $M$ is a non-standard model of complete arithmetic, i.e. of the theory $Th(\mathbb{N})$. Suppose that $R$ and $S$ are proper cuts of $M$. (With a cut, I mean a subset of the universe of ...

**5**

votes

**1**answer

238 views

### Scott sentence in models of set theory

Let $\mathfrak{M}$ be a countable transitive model of set theory.
Let $L$ be some countable language and $A$ be a countable (in $\mathfrak{M}$) $L$-structure.
My question is:
In $\mathfrak{M}$ can ...

**7**

votes

**1**answer

240 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 ...

**1**

vote

**0**answers

65 views

### Is this dichotomy for a VC-shatter-like function known?

Associated with each family $\mathcal{F}$ of a subsets of a set $\mathbf{X}$ I have a function $f:\mathbb{N}\rightarrow\mathbb{N}$ resembling the dual shatter function. This function obeys a ...

**3**

votes

**2**answers

224 views

### Sufficient Condition for Defining $\in$

Consider the first order language $\mathcal{L}=\{\in,\in'\}$ with two binary relational symbols $\in , \in'$ and $ZFC$ as a $\{\in\}$-theory. If we define $\in'$ using $\{\in\}$-formula $\varphi(x,y)$ ...

**1**

vote

**1**answer

227 views

### A Special Pair of Formulas

Consider the first order language $\mathcal{L}=\{\in,\subseteq\}$ and $\{\in\}$-theory $\text{ZFC}$. Is there a formula $\psi (x,y) \in \{\subseteq\}-Form$ with the following ...

**1**

vote

**1**answer

114 views

### infinitary logic and partial fixed point logic

Is there a property definable in finite-variable infinitary logic $L^{\omega}_{\omega\infty}$ but not definable in partial fixed point logic FO(PFP) ?

**4**

votes

**1**answer

235 views

### Quantifier elimination vs decidability

Quantifier elimination is used as a technique to get decidability (e.g. $Th( \mathbb{N}, +)$ ) of theories, but typically one has to go over to some expansion. Are there examples of theories which are ...

**6**

votes

**3**answers

378 views

### Systematic brute-force searches for counterexamples

This is getting nowhere on math.stackexchange.com, so I'm putting it here.
Gödel's completeness theorem says that for every statement in first-order predicate calculus with equality, there is either ...

**9**

votes

**2**answers

507 views

### What is the precise notion of “enough arithmetic” in Godel's first Incompleteness theorem?

I'm trying to reconstruct the proof of Godel's first theorem (Rosser's strong version) from the uncomputability of the Halting function. If we just started with the language $\mathcal{L}=\{0, S, +, ...

**8**

votes

**2**answers

361 views

### The (non-)absoluteness of second-order elementary equivalence

Elementary equivalence is set-theoretically absolute between any two transitive models of set theory; this is also true for the infinitary logics - e.g., $\mathcal{L}_{\omega_1\omega}$ - at least, ...

**1**

vote

**1**answer

105 views

### Preserving Predimension Functions under Functional Convergences

Definition 1. If $\mathcal{L}$ is a countable relational language, a predimension class $C$ is a class of $\mathcal{L}$-structures with the following properties:
C1: ...

**2**

votes

**0**answers

126 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 ...

**4**

votes

**1**answer

174 views

### Stable examples from Algebra such that the model theoretic algebraic closure of a substructre is no model

Let $T$ be a stable theory. Let $A$ be a subset or substructure of a model $M$ of $T$. Now in some theories the (model theoretic) algebraic closure of $A$ is already a (sub)model of $T$. For example, ...

**4**

votes

**0**answers

134 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.
...

**9**

votes

**3**answers

387 views

### Is a model of arithmetic contained in a model of arithmetic an initial segment?

It's easy enough to show that if $\mathbb{N}_1$ is a non-standard model of the Peano axioms, then there is a canonical embedding $\mathbb{N} \to \mathbb{N}_1$, and we have a theorem that if $x \in ...