**2**

votes

**1**answer

108 views

### Precise interpretability strength of $\mathcal P_{DF}(\omega)$ and $L_{\omega_1^{CK}}$

I am curious about the relationship between the definable power set of $\omega$ and the $\omega_1^{CK}$th level of the constructible sets $L$.
In short, $\omega_1^{CK}$ is the least nonrecursive ...

**2**

votes

**0**answers

80 views

### Is the dg-nerve functor a Quillen equivalence?

Lurie defines the dg-nerve $N_{dg}(\mathcal{C})$ of a dg-category $\mathcal{C}$ in Higher Algebra Construction 1.3.1.6: for each $n \geq 0$, we define $N_{dg}(\mathcal{C})_n\simeq ...

**3**

votes

**1**answer

209 views

### History of unstable formulas [closed]

There are many equivalent definitions for stability, one of them being that being unstable is equivalent to the existence of a formula having the order property.
While intuitively it makes sense that ...

**3**

votes

**1**answer

194 views

### Löwenheim-Skolem for many-sorted theories

Let $L$ be a many-sorted first order language, and let $\kappa$ be an infinite cardinal which is greater than or equal to the number of function and relation symbols in $L$. Let $T$ be a complete ...

**0**

votes

**0**answers

118 views

### Minimum regular open set containing a given set in a T0 Alexandrov topological space

What is known about the minimum regular open set containing a given set in a T$_0$ Alexandrov topological space? I'm particularly interested in the condition for the minimum set happening to be ...

**5**

votes

**1**answer

142 views

### Do models-and-homomorphisms always form an accessible category?

It's well-known that the category of models of any first-order theory $T$ form an accessible category if we take the elementary embeddings as morphisms. This is true in finitary first-order logic or ...

**1**

vote

**0**answers

72 views

### Nice model theoretic properties of a theory after adding predicates

I would like to know what nice model theoretic properties (for example simplicity, NIP, stability, etc) can be preserved when we add a new predicate to the language.
Explicitly, if T is an L-theory ...

**2**

votes

**1**answer

185 views

### Classification Theory

It is often said that unsuperstable theories do not admit a classification in the sense of Shelah. Why exactly is this so? And also what exactly does in the sense of Shelah mean? It is hand waved in a ...

**3**

votes

**1**answer

172 views

### Number of non-isomorphic models

I had this question up on Math stackexchange: http://math.stackexchange.com/questions/1349247/number-of-non-isomorphic-models/1350763#1350763 . While it was answered partially there, I'm posting here ...

**3**

votes

**1**answer

128 views

### Non-Forking and Related Concepts

Is the importance of developing forking machinery in the way we set it up, or is it in the fact that it allows us to come up with a notion of independence via the properties of non-forking? I'm ...

**5**

votes

**0**answers

88 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 ${\cal M}$ ...

**0**

votes

**1**answer

211 views

### Provability of unprovability

I have three questions (without any real background, this is just something I've been wondering about recently)
Can PA prove that PA can't prove nor disprove, say, Goodstein theorem (or any natural ...

**5**

votes

**0**answers

95 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 ...

**2**

votes

**1**answer

101 views

### Epsilon-approximations of set systems with finite VC-dimension

ECorollary 6.9 in A Guide to NIP theories by Pierre Simon proves the following
Theorem. For every positive integer $k$ and every positive real $\varepsilon$ there is an integer $n=n(k,\epsilon)$ ...

**5**

votes

**1**answer

229 views

### Looking for reference or proof to some facts stated on Anand Pillay's book

In my current work I am using facts 2.1.11 and 2.1.12 from Anand Pillay's book Geometric Stability Theory.
The facts are stated as follows:
Fact 2.1.11. Let $(S,\mbox{cl})$ be a locally ...

**6**

votes

**1**answer

289 views

### If two structures are elementarily equivalent, is there a zigzag of elementary embeddings between them?

Fix a first-order signature $\Sigma$. There is an equivalence relation $\sim$ on the class $\Sigma\mathrm{-Str}$ of all $\Sigma$-structures given by $M \sim N$ iff $M$ and $N$ are elementarily ...

**1**

vote

**1**answer

302 views

### Is second-order ZFC categorical with regard to its proper class models

Second-order ZFC offers partial categoricity in the sense that, given any two models, one of them must be isomorphic to an initial segment of the other [1]. However, this leaves questions regarding ...

**2**

votes

**1**answer

192 views

### Is there a 'largest' second-order categorical axiomatization of set theory, extended from ZFC2

While it's possible to obtain categorical second-order axiomatizations of set theory by extending ZFC2 with additional axioms (see [1]), these axioms tend to be somewhat arbitrary (e.g. adding an ...

**4**

votes

**0**answers

117 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 ...

**17**

votes

**4**answers

689 views

### Is there a Leibnizian model with no definable elements, in a finite language?

A first-order structure $M$ is Leibnizian, if any two distinct
elements $a,b\in M$ satisfy different $1$-types; that is, if there
is some formula $\varphi$ such that $M\models\varphi(a)$ and
...

**8**

votes

**2**answers

301 views

### Around Vopěnka: Accessible category with small full discrete subcategories of arbitrary size?

I believe the model-theoretic version of the question is: is there a theory in finitary first-order logic which has, for each cardinal $\lambda$, a set $C_\lambda$ of $\lambda$-many models, such that ...

**1**

vote

**0**answers

125 views

### What can be said about a Boolean-valued structure from what the Boolean-valued forcing extension thinks about it?

Suppose that $\phi$ is a formula in the language of set theory such that
there are some $n_{1},...,n_{k}$ such that if $V\models\phi(x)$, then $x=(X,R_{1},...,R_{k})$ and ...

**13**

votes

**2**answers

906 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 ...

**5**

votes

**1**answer

101 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 ...

**9**

votes

**1**answer

102 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 ...

**5**

votes

**0**answers

183 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 ...

**1**

vote

**0**answers

100 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

101 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 ...

**2**

votes

**1**answer

208 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

332 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

68 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

103 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 ...

**2**

votes

**0**answers

174 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 ...

**1**

vote

**1**answer

127 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 ...

**8**

votes

**3**answers

313 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 ...

**16**

votes

**6**answers

883 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 ...

**6**

votes

**0**answers

89 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 ...

**13**

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

68 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 $E$ ...

**8**

votes

**1**answer

189 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 ...

**6**

votes

**1**answer

131 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

398 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

**2**answers

890 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 ...

**-1**

votes

**1**answer

106 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

210 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 $P(\omega)$ ...

**11**

votes

**1**answer

565 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 ...

**3**

votes

**1**answer

137 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

180 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 ...

**16**

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 ...

**10**

votes

**2**answers

449 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 ...