**3**

votes

**4**answers

337 views

### A Fraïssé class without the strong amalgamation property.

I am trying to find some examples of Fraïssé classes that do not have the strong amalgamation property. Anyone?

**6**

votes

**0**answers

59 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 ...

**4**

votes

**1**answer

145 views

### Uncountably categorical theories which are interpretable in a strongly minimal

Definition: Let $\lambda$ be a cardinal. An $\mathcal{L}$-theory $T$ is called $\lambda$-categorical whenever every two models of $T$ of cardinality $\lambda$ are isomorphic.
Definition: An ...

**11**

votes

**2**answers

1k views

### Do the real numbers “know” that they are countable in a larger model?

(This was first posted to math.stackexchange but had no answers there after several days):
Let ${\mathbb R}$ be the set of real numbers in whatever is your favorite model of $ZFC$. Then (by Levy ...

**2**

votes

**0**answers

53 views

### Potentiality classes and Borel reductions

In a 1998 paper by Hjorth, Kechris, and Louveau, there was a definition given of a "potentiality class." That is, given an invariant equivalence relation $E$ on a standard Borel space $X$, we say $E$ ...

**6**

votes

**1**answer

94 views

### Chains of forking extension in stable theories

Let $T$ be an stable theory.
Further we work in the monster model of $T^{eq}$.
We say that a chain of types of the form
$$tp(a_1/A_1)\subset tp(a_2/A_2) ... \subset tp(a_n/A_n)$$
is a forking chain ...

**-4**

votes

**0**answers

77 views

### every(ultra)filter on set I is principle if and only if I is finite [closed]

1)the filter generated by{a,b} is not ultra filter?
2)the filters generated by singleton are precisely the principle ultrafilters.
3)every(ultra)filter on set I is principle if and only if I is ...

**-2**

votes

**0**answers

30 views

### Underlying Set in Model Theory [migrated]

In model theory a structure has an underlying set.
In addition to the interpreted relations, are there
(implicit) assumptions made about possible
operations on this set? For example, is it assumed to ...

**7**

votes

**1**answer

215 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

**1**answer

346 views

### Alternate proof of Morley's theorem?

I'm trying to understand the result given in the first box at slide 45 of this talk. Specifically:
1) What is the source cited? I have not been able to find any article by Keisler, Chudnovsky and/or ...

**13**

votes

**3**answers

652 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 ...

**3**

votes

**2**answers

549 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 ...

**12**

votes

**2**answers

745 views

### nonstandard models and mathematical theorems

Is there a first order statement about the natural numbers (not nonstandard analysis) such that the truth of the statement is easier to see in a nonstandard model? In other words, do nonstandard ...

**5**

votes

**2**answers

412 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 ...

**6**

votes

**1**answer

1k views

### Definable measure preserving isomorphisms of $p$-adic semialgebraic sets

Hi,
Consider a $p$-adic field $K$ (finite extension $\DeclareMathOperator{\bQ}{\mathbb{Q}}$of $\bQ_p$) in Macintyre language $\DeclareMathOperator{\cL}{\mathcal{L}}$ $\cL_{\rm Mac}$. Let $Z$ be a ...

**-1**

votes

**1**answer

95 views

### types and elementary extensions [closed]

Let $\mathcal{M}$ and $\mathcal{N}$ be two $\mathcal{L}$-structures and suppose that for n-tupls $\bar{a}\in M^n$ and $\bar{b}\in N^n$,
$tp^\mathcal{M}(\bar{a})=tp^\mathcal{N}(\bar{b})$ where ...

**8**

votes

**0**answers

196 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 $P(\omega)$ ...

**2**

votes

**0**answers

69 views

### Uniqueness of an embedding theorem for Real differential fields

I will follow a preliminary exposition for the problem in question, which will essentially follow the format on http://www4.ncsu.edu/~singer/papers/model_diff_fields.pdf [pg. 87]:
Let $K$ be a real ...

**10**

votes

**1**answer

515 views

### Is forcing computable?

By results similar to Tennenbaum's theorem we know that there exist no computable models of $ZF$. But suppose we are given, as a sort of oracle, access to some model of $ZF$ (e.g. we can make oracle ...

**-1**

votes

**1**answer

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 ...

**15**

votes

**2**answers

1k views

### Is non-existence of the hyperreals consistent with ZF?

I know that it is possible to construct the hyperreal number system in ZFC by using the axiom of choice to obtain a non-principal ultrafilter. Would the non-existence of a set of hyperreals be ...

**3**

votes

**1**answer

123 views

### Formal systems needed to formalize relative independence results

We know that Con(ZF) implies Con(ZFC+GCH), Con(ZF+neg(AC)) and Con(ZFC+neg(CH)). But what are some weak theories in which these relative independence results are provable? In particular, are they ...

**6**

votes

**2**answers

174 views

### Formal languages with non-unique interpretations of terms

In mathematical logic and model theory, one considers interpretations of syntactic expressions: terms without free variables are interpreted as elements of some structure, formulas without free ...

**10**

votes

**2**answers

436 views

### Can a parent and child node have the same type in a well-founded digraph tree?

$\newcommand\toward{\rightharpoonup}$It would help me to
understand something in a current research project if someone
could provide an example of directed graph $\langle
G,\toward\rangle$ with the ...

**1**

vote

**1**answer

104 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: ...

**6**

votes

**1**answer

97 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

460 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

313 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

189 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

434 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

90 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 $X$ ...

**22**

votes

**7**answers

2k views

### Supervenience in mathematics

I'm not quite sure if this is the right place to ask, and if this is the right way to ask, but I dare.
In philosophy (of mind, e.g.) the concept of supervenience is used:
"Supervenience [is] used ...

**1**

vote

**1**answer

101 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

167 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 ...

**7**

votes

**1**answer

350 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 ...

**1**

vote

**0**answers

80 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 ...

**21**

votes

**2**answers

1k 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 ...

**7**

votes

**1**answer

259 views

### Vaught conjecture for uncountable languages

Recall Vaught conjecture: the number of countable models of a first-order complete theory in a countable language is finite or $\aleph_0$ or $2^{\aleph_0}.$
Now let $\lambda$ be an uncountable ...

**12**

votes

**0**answers

759 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 ...

**6**

votes

**3**answers

339 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

**2**answers

623 views

### Are there standard examples of stable theories that are undecidable?

What is known about decidability of various first order theories studied in stable model theory, geometric model theory, o-minimality? For example, is there a natural example of an undecidable ...

**9**

votes

**1**answer

564 views

### an algebraically closed field definable in a real closed field

Is it true that an algebraically closed field $k$ definable (in the model theory sense) in a real closed field $\mathcal R$ is the algebraic closure $\mathcal{R}^{alg}=\mathcal{R}(\sqrt{-1})$?
...

**4**

votes

**1**answer

261 views

### A totally categorical structure with trivial geometry which is not interpretable in the trivial structure

Among the theorems of early geometric model theory there is one by Lachlan stating that every totally categorical structure with a trivial pregeometry is intrepretable in a dense linear order.
That ...

**7**

votes

**1**answer

150 views

### Definability of Morley rank in the theory of Compact Complex Spaces(CCS)

Is there a counter-example showing that Morley rank in the theory of compact complex spaces (as defined by Zilber, Pillay, Moosa, ...) is not definable in families?
Given the existence of such a ...

**14**

votes

**5**answers

777 views

### What is the spectrum of possible cofinality types for cuts in an ordered field? Or in a model of the hyperreals? Or in a nonstandard model of arithmetic?

I am interested to know the full range of possibilities for the cofinality type of cuts in an ordered field and in other structures, such as nonstandard models of arithmetic.
Definitions. ...

**5**

votes

**0**answers

212 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.
...

**2**

votes

**0**answers

55 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 ...

**5**

votes

**3**answers

290 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

136 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, ...

**8**

votes

**1**answer

287 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 ...