**5**

votes

**1**answer

64 views

### When is a formula preserved under taking factors in a reduced product or the stalk in a Boolean product?

I want to know if there is a nice characterization of when a formula is preserved under taking reduced factors.
We say that a formula $\phi$ is closed under taking reduced factors if whenever $I$ is ...

**5**

votes

**0**answers

152 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 ...

**11**

votes

**2**answers

791 views

### When does Vopěnka's principle hold?

Vopěnka's principle (VP) is the statement that, given any proper class $\{\mathcal{A}_\eta: \eta\in ON\}$ of first-order structures in the same language, there are some $\alpha\not=\beta$ with ...

**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 ...

**5**

votes

**1**answer

70 views

### Strongly minimal set with DMP

Recall that a strongly minimal theory $T$ has the Definable Multiplicity Property (DMP) if for all natural $k$, $m$ and $\varphi(\bar{x},\bar{b})$ of rank $k$, multiplicity $m$, there exists a formula ...

**16**

votes

**5**answers

2k views

### Category theory and model theory as “natural” counterparts

I am aware of the profound discussion of the relationship between category theory and model theory (e.g. at The n-Category Café) but do wonder why category theory (CT) is not opposed to model theory ...

**1**

vote

**0**answers

76 views

### saturated model [closed]

Suppose you have a saturated model N of a complete theory T without finite models. How is it possibile to construct a proper saturated elementary substructure of N of the same cardinality of N ?
I ...

**3**

votes

**0**answers

74 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 ...

**13**

votes

**6**answers

811 views

### Application of Fraïssé construction in set theory

As you know Fraïssé limit construction and its generalization, Hrushovski's construction, have many applications in model theory to build models with interesting property.
Now I would like to know ...

**2**

votes

**1**answer

191 views

### Mal'cev “rational equivalence” and model theory

In Universal Algebra, it is possible to say that two presentations denote the "same" kind of algebraic structures, if the two corresponding varieties are "rationally equivalent" (Mal'cev 1958).
In ...

**4**

votes

**1**answer

288 views

### Does PA+Con(PA) entail the existence of non-standard models of PA?

Does $\textsf{PA}$+Con($\textsf{PA}$) entail the existence of non-standard models of $\textsf{PA}$?
Is there a reasonable way in which to code, inside $\textsf{PA}$, the statement that $\textsf{PA}$ ...

**1**

vote

**0**answers

65 views

### Counting models in first order logics without existencial quantifiers

My question is about the posibility of to construct a parameter space of models in a first order theory, finitely presented, with out existencial quantifiers (parameter space in the sense of ...

**2**

votes

**1**answer

86 views

### Representable cylindric algebras and correspondence with first-order models

The class $\textsf{Cs}_{\omega}^{reg}\cap \textsf{Lf}_{\omega}$ of locally finite and regular cylindric set algebras (of dimension $\omega$) can be seen as the algebraic counterpart of first-order ...

**12**

votes

**1**answer

377 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 ...

**36**

votes

**7**answers

3k views

### Is the ultraproduct concept fundamentally category-theoretic?

Once again, I would like to take advantage of the large number of knowledgable category theorists on this site for a question I have about category-theoretic aspects of a fundamental logic concept.
...

**2**

votes

**0**answers

163 views

### Automorphisms of $\mathbb{C}$ and meromorphic functions

Let $F$ be a meromorphic function on $\mathbb{C}$, and assume that the first-order theory of $(\mathbb{C},F)$ defines $\mathbb{Z}$, which means that there exists a formula $\varphi(z)$ (in the ...

**7**

votes

**1**answer

236 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

**1**answer

107 views

### Relationship between a first-order theory and its model companion

Given a first-order theory $T$ its model companion $T^*$ can be considered. Under relatively weak assumptions on $T$ we have that $T^*$ is the theory of the class of the existentially closed models ...

**7**

votes

**6**answers

501 views

### Applications of forcing in model theory

What are the major applications of (set theoretic) forcing in model theory?

**8**

votes

**3**answers

276 views

### Model-theoretic accounts of feasibility in bounded arithmetic and related systems

Various weak theories of arithmetic have been partially motivated by a concern with numbers (or functions/proofs) that are feasible. This concern is sometimes connected to an interest in strictly ...

**3**

votes

**4**answers

352 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

80 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 ...

**6**

votes

**1**answer

172 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 ...

**12**

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

64 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

119 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 ...

**6**

votes

**1**answer

382 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

663 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

555 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 ...

**13**

votes

**2**answers

823 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

421 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

98 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

201 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

72 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

536 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

303 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

130 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

178 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

447 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

107 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

487 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

322 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

198 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

447 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

93 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

107 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 ...