**4**

votes

**1**answer

183 views

### Replacement axiom and existence of end-extensions (Chang and Keisler's Model Theory)

I am currently reading Chang and Keisler's model theory textbook, and there is something that I don't seem to be able to understand about their proof (through the Omitting Types Theorem) that every ...

**6**

votes

**1**answer

778 views

### Is the first-order theory (with =) of real numbers with addition and multiplication complete and decidable?

Due to Andreas Blass's answer to my question "Is the feasibility of a system of nonlinear, non-convex equations (inequalities) decidable?", i have now investigated real closed fields (RCF), because i ...

**1**

vote

**0**answers

149 views

### Self-modelling structures

Consider - for the sake of simplicity - only graphs as structures.
For undirected graphs $(V, E\subseteq \binom{V}{2})$ let
$E(v)$ be the set of edges $e\in E$ incident with $v$, i.e. $\lbrace e ...

**2**

votes

**1**answer

205 views

### Every abelian torsion-free group is strictly totally orderable (via the compactness theorem)

Let $\mathbb G = (G, +)$ be a group. We say that $\mathbb G$ is strictly totally orderable (others would say bi-orderable) if there exists a total order $\preceq$ on $G$ such that $x+z \prec y + z$ ...

**3**

votes

**0**answers

161 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, ...

**1**

vote

**1**answer

146 views

### Importance of Denjoy-Carleman classes as a class.

Denjoy-Carleman classes of differentiable functions, say in Roumieu's form:
Given a log-convex sequence $M_n$ of positive number denote by $C_M=C_M(\mathbb{R}^n,0)$ the ring of germs of ...

**2**

votes

**1**answer

166 views

### dp-minimality and stability

What are some of the common popular stable theories that are known to be dp-minimal (or not dp-minimal)?
Some dp-minimal examples I am aware of are strongly minimal theories, superstable theories of ...

**1**

vote

**1**answer

212 views

### Elementary extensions and type spaces

If $M$ and $N$ are two $L$-structures, and $f: M \rightarrow N$ is an elementary extension, then given any subset $A$ of $M$, $f$ induces in a natural way a morphism $S^M_n(A) \rightarrow S^N_n(f(A))$ ...

**3**

votes

**2**answers

196 views

### Validity in Kripke frames whose points are finite or infinite sequences

Suppose $D$ is a non-empty set and $\{ R_i : i \in \mathbb{N} \}$ is a family of binary relations on sequences over $D$ so that $R_i \subseteq D^i \times D^i$. Let $R_\omega \subseteq D^\omega \times ...

**11**

votes

**1**answer

435 views

### First order decidability of rings vs Diophantine decidability

Are there known (preferably ``concrete'') examples of a ring $R$ (commutative, with 1) such that:
$\bullet$ the first order theory of $R$ is undecidable, but
$\bullet$ the positive existential (= ...

**1**

vote

**2**answers

136 views

### Approximating a function via definable functions II

In a previous post I asked about the definability of a function that can be "approximated" by a uniformly definable family of functions. Nevertheless, the notion of approximation I gave was too weak ...

**1**

vote

**1**answer

171 views

### Approximating a function via definable functions

Let $T$ be a first order theory, $M$ a model of $T$ equipped with a topology with a definable basis (i.e. every basic open is definable with parameters). Let $F: M\rightarrow M$ be a partial function ...

**3**

votes

**3**answers

280 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?

**2**

votes

**1**answer

196 views

### Definability in a language with a single binary predicate

Let the first-order language ${\mathcal{L}}$ have a single binary predicate $P$. Consider the structure whose underlying set is ${\mathbb{Z}}$, the integers, and an ordered pair $(m,n)$ is in $P$ if ...

**3**

votes

**1**answer

255 views

### Barwise compactness theorem

In Admissible Sets and Structures, page 101, theorem 5.8, Barwise introduces a weird form of his compactness theorem in which there are two theories $T$ and $T'$ both $\Sigma_1$, such that every ...

**1**

vote

**1**answer

815 views

### Are there non-commutative models of arithmetic which have a prime number structure?

Peano Arithmetic (PA) models have a prime number structure and commutativity of addition and multiplication. Presburger arithmetic (PrA) models of arithmetic have addition without multiplication and ...

**1**

vote

**1**answer

224 views

### A (seem to be) elementary logic question

Given language $L$. $P$ is a 1-place predicate in $L$. Let language $L_0 = L \setminus \{P\}$. Let $\sigma$ be a sentence of $L$ (may contain symbol $P$). $\mathfrak{A}$ is a structure of $L$, and ...

**2**

votes

**1**answer

437 views

### What follows from assuming not Con(ZF)?

Hello.
Let $\operatorname{sat} X$ denote the satisfiability of a theory $X$.
From Gödel's second incompleteness theorem and his completeness theorem follows $$ZF \not\vdash \lceil \operatorname{sat} ...

**1**

vote

**0**answers

158 views

### What are the enforceable models of local artinian rings?

I was reading Hodges' "Model Theory" Chapter 8 a propos existentially closed models of $\forall_2$ theories in a countable first order language $L$. He extends the proof of the omitting type theorem ...

**0**

votes

**0**answers

283 views

### Definitions for Oddness

In the thread Even Xor Odd Infinities I defined odd models of Modular Arithmetic (MA) as models satisfying the axioms of MA and two first order statements. Even XOR Odd Infinities?
$O1) \forall x(x=0 ...

**5**

votes

**1**answer

265 views

### Is there any o-minimal expansion of the real field with functions of growth higher than exponential?

Let $\bar{\mathbb{R}}$ be the structure of the real field, that is $(\mathbb{R},0,1,+,-,*,<)$ . We say that a function $f$ is of growth higher than exponential if for all $N\in \mathbb{N}$ there ...

**8**

votes

**0**answers

298 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). ...

**4**

votes

**0**answers

219 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

**1**answer

164 views

### When do substructures have computable copies?

Say that a class $\mathcal{C}$ of countable first-order structures in some finite signature has the effective substructure property if $\mathcal{C}$ is closed under isomorphism and whenever $A\in ...

**1**

vote

**0**answers

105 views

### Compactness-like property for universal generalization?

Hi all! I have the following problem. Suppose I have a sequence of models $M_1,M_2,...$, all of which have the same countable domain (call it $D$). $x_1,x_2,...$ is a well-ordering of $D$. $\phi(x)$ ...

**10**

votes

**2**answers

430 views

### Constructible models of New Foundations?

Hi all! Is there anything like Gödel's constructible universe for New Foundations?
More precisely, I would like a process for taking a model $M$ of NF, and using it to build a model $L \subseteq M$ ...

**4**

votes

**1**answer

215 views

### tennenbaum phenomena for the reals?

Let $\mathfrak{M} = \langle R, +,\times,> \rangle$ be such that $R$ is the set of real numbers and $\mathfrak{M} \models RA^1$ (the first-order axioms for the reals). Do we have characterisations ...

**2**

votes

**0**answers

211 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 ...

**3**

votes

**1**answer

273 views

### Why Ryll-Nardzewski theorem fails for uncountable theories?

Ryll-Nardzewski theorems states that if $T$ is a countable complete theory, then $T$ is $\aleph_0$-categorical if and only if for every $n<\omega$ there are only finitely many formulas ...

**1**

vote

**1**answer

124 views

### galois correspondence for hyperimaginaries

If there is a hyperimaginary $a_{E}$ and sets: $A,B\subset M$,
such that for every automorphism F which fix A , F fix $a_{E}$,
and for every automorphism F which fix B , F fix $a_{E}$.
Does it true ...

**6**

votes

**1**answer

250 views

### Non-definable elements vs indiscernible elements

Let $\Sigma$ be a one-sorted first-order signature, let $A$ be a $\Sigma$-structure, and let $B \subseteq A$ be a $\Sigma$-substructure. Fix a class $\mathcal{L}$ of formulae over $\Sigma$. We say an ...

**4**

votes

**1**answer

157 views

### $L_{\omega_1,\omega}$ sentence with many automorphism in $\aleph_0$ and few automorphism in $\aleph_\omega$

Hello,
I have the following question: Is it possible for a complete $L_{\omega_1,\omega}$ sentence $\phi$ to satisfy (a) the (unique) countable model of $\phi$ has $2^{\aleph_0}$ many automorphisms ...

**2**

votes

**1**answer

348 views

### Decidability survives new constants

Let $L$ be a finite first order language
and let $M$ be an $L$-structure with universe $\mathbb{N}$
that interprets all $L$-symbols as recursive sets
(so $M$ is a recursive $L$-structure).
Let ...

**6**

votes

**1**answer

424 views

### (Finite) Models of two subtheories of Peano Arithmetic

Consider first-order theory (with identity) of Peano Artithmetic built in the language $\{S,+,\times,0\}$ and with the following set of axioms:
\begin{align}
\neg Sx&=0\tag{1}\\\
...

**5**

votes

**1**answer

330 views

### Smallest real closed field realizing all cuts of the rational numbers

Let $K$ be a real closed field of transcendence degree 1 over $\mathbb{R}$.
It is not difficult to see that $K$ has the following "minimality property": Whenever $L$ is a real closed field that ...

**10**

votes

**1**answer

359 views

### Number of linear orders

It is well known that for every infinite cardinal $\kappa$ the number of non-isomorphic total orders of cardinality $\kappa$ is $2^\kappa$. Who first proved this, and in what context? Was it proved ...

**4**

votes

**3**answers

583 views

### Survey of finite axiomatizability for relational theories?

An $L$-theory $T$ is finitely axiomatizable if there is a finite set $A$ of $L$-sentences with the same consequences as $T$, i.e. such that $M \models T$ iff $M \models A$ for every $L$-structure $M$. ...

**3**

votes

**1**answer

196 views

### Nondegenerate involutions on the complex numbers — always just conjugation?

Suppose you have a ring homomorphism $(-)':\mathbb{C} \to \mathbb{C}$, which is an involution such that $\sum_i a_i a_i' = 0 \Leftrightarrow \forall i \ a_i=0$, where this sum is finite.
Must this ...

**2**

votes

**3**answers

338 views

### Methods for proving non FO definability

I'm having difficulties to prove that the subset of even numbers is not first-order definable in $(\mathbb{N},<)$. Any hint is welcomed!
More generally, what are usual techniques in order to prove ...

**0**

votes

**1**answer

279 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 ...

**1**

vote

**1**answer

184 views

### Existentially closed substructure and ultraproducts

Is it true that if M is existentially closed in N then N can be embedded in an ultraproduct of M ?
Best regards.

**2**

votes

**2**answers

324 views

### Applications of the Ax Kochen Ershov (AKE) princicple

The AKE priciple states that two finitely ramified Henselian field (this means that the field is either of residual caracteristic 0 or is of characteristic $p$ and there is only a finite number of ...

**0**

votes

**1**answer

155 views

### Nonsingular zeroes are algebraic?

I'm getting started in Real Algebraic Geometry (from a model-theory perspective), and a paper makes the following assertion (here $K\subset L$ are real-closed fields):
Suppose $Q\in L^n$, $f_1, ...

**1**

vote

**2**answers

315 views

### How do we avoid circularity when we build a structure for ZFC? [closed]

when investigating ZFC as a formal language a structure is a set, are we not engaging in circular logic here? Or is 'set' thought of in a more primitive sense?

**9**

votes

**2**answers

672 views

### Intuition behind o-minimal structures.

This is very much the same post as I posted at math.stackexchange.
I am following the definitions of an o-minimal system in "Tame Topology and O-minimal Structures" by Lou van den Dries.
It is ...

**1**

vote

**0**answers

167 views

### Characterizing $\mathbb{Q}$ among number fields

Is there an $\mathcal{L}_{\omega_1\omega}$ formula or set of formulas that characterizes the rationals $\mathbb{Q}$ among other number fields?
EDIT: My formula must not contain an infinite number of ...

**2**

votes

**0**answers

159 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 ...

**3**

votes

**1**answer

353 views

### Interpretations as morphisms

Model-theoretic interpretations seem to give rise to categories in which morphisms are not functions. To name just two small examples:
the category of finite graphs with interpretations between them
...

**6**

votes

**1**answer

791 views

### Where do nonstandard elliptic curve angles come from?

This is a question which has bounced around my head over the past few years. At the same time, I am answering ...

**3**

votes

**2**answers

324 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 ...