**4**

votes

**1**answer

141 views

### Sketches for categories of models of complete theories

In Accessible categories : the foundations of categorical model theory, chapter 3 p.58, Makkai and Paré claim that there is "an (obvious) identification of a class of sketches so that the categories ...

**3**

votes

**1**answer

168 views

### Completeness theorem via syntactic categories

The nLab says in its internal logic article that the Completeness Theorem can be proven via a ``generic model'' of the theory. The model is generic in the sense that the only things true of it are ...

**3**

votes

**2**answers

442 views

### Are descriptive and ontological notions of equality equal? [closed]

Let $a$ and $b$ are two "objects". What is the meaning of $a=b$? This is one of the deepest problems of philosophy and logic because one needs a complete information about ...

**8**

votes

**1**answer

314 views

### Does existence of a proper class model imply the consistency?

The fundamental theorem of model theory says that:
Theorem: A first order theory is consistent if and only if it has a model.
In the above theorem we assume that the domain of any model is a ...

**4**

votes

**2**answers

278 views

### Can we force with Fraisse filters to solve Vaught's conjecture?

Around the classic Fraisse amalgamation theorem in model theory we have the following notions:
Definition (1): If $M$ be an $\mathcal{L}$-structure then define:
$age(M):=\lbrace N~|~N~\text{is ...

**1**

vote

**0**answers

154 views

### Is there a non-trivial consistency preserving transformation?

In set theory "equiconsistency" (and not "consistency") of the theories is the main part of researches. So we usually try to construct a new model using a given one. In the ...

**5**

votes

**1**answer

174 views

### What are the Possible Large Cardinals of $L[X]$?

I've been doing some basic reading in inner model theory, and I'm at the point where I've seen the definition of things like Martin-Steel and Mitchell-Steel inner models. I am wondering about the ...

**7**

votes

**1**answer

387 views

### Shelah's categoricity conjecture

Does Shelah's categoricity conjecture for abstract elementary classes have applications in other branches of mathematics?

**12**

votes

**3**answers

761 views

### Where is the end of universe?

In some sense the empty set ($\emptyset$) and the global set of all sets ($G$) are the ends of the universe of mathematical objects. The world which $ZFC$ describes has an end from the bottom and is ...

**5**

votes

**1**answer

165 views

### Absoluteness of completeness

Suppose $V_0, V_1$ are (not necessarily well-founded) models of ZFC and suppose $\varphi$ is a first order sentence in a finite language $L$ (in our background model of set theory). Because every true ...

**7**

votes

**1**answer

184 views

### Is there a forcing closure?

The main theorem of forcing says that for any c.t.m of $ZFC$ like $M$ and for all partial order $\mathbb{P}$ and $\mathbb{P}$-generic $G$ over $M$, there is a c.t.m of $ZFC$, like $N$ such that $N$ is ...

**9**

votes

**1**answer

275 views

### Are all complete finitely axiomatizable first order theories $\aleph_0$-categorical?

Suppose $T$ is a complete first order theory with a finite axiomatization. Must $T$ be $\aleph_0$-categorical? If not are there any simple examples of finitely axiomatized complete first order ...

**1**

vote

**0**answers

163 views

### Relativization of Formulas and Models [closed]

I want to show that the definition of satisfiability is consistent with the definition given by relativization, i.e. Let $L=\{\in\}$. Let $M$ be a definable set and let $E\subset{M\times{M}}$. Let ...

**3**

votes

**1**answer

178 views

### Do inner models of unique measurable cardinals have a regular behavior? (Edited and Revised Version)

We know that if $\kappa$ is a measurable cardinal and $\mu$ be a two-valued non-trivial$\kappa$-additive measure on it then the corresponding inner model produced by ...

**5**

votes

**2**answers

223 views

### Is there an inner model between two distinct inner models of ZFC?

Definition (1): An inner model of $ZFC$ is a tarnsitive proper class model of $ZFC$ which contains all ordinal numbers. Informally we denote the collection of all inner ...

**22**

votes

**4**answers

1k views

### Illustrating Edward Nelson's Worldview with Nonstandard Models of Arithmetic

Mathematician Edward Nelson is known for his extreme views on the foundations of mathematics, variously described as "ultrafintism" or "strict finitism" (Nelson's preferred term), which came into the ...

**5**

votes

**1**answer

281 views

### What are the essential properties of algebraic closure on an arbitrary structure?

Define the "model theoretic" notion of a closure function as follows:
Definition (1): Let $D$ be a non-empty set. A function $cl:P(D)\longrightarrow P(D)$ called a closure function iff it has the ...

**23**

votes

**3**answers

516 views

### Is the field of constructible numbers known to be decidable?

By the field of constructible numbers I mean the union of all finite towers of real quadratic extensions beginning with $\mathbb{Q}$. By decidable I mean the set of first order truths in this field, ...

**2**

votes

**1**answer

195 views

### Can we flex the rigid models by enough power?

Definition (1): An $\mathcal{L}$ - structure $\mathcal{M}$ called "rigid" iff there is no non-trivial automorphism on $\mathcal{M}$.
Definition (2): An ...

**4**

votes

**1**answer

241 views

### Omitting types and Baire category

What is the relation between omitting types theorems in model theory and the baire category theorem?

**9**

votes

**2**answers

294 views

### Extending a partial order while preserving an automorphism

It is well known that if $(P, \leq)$ is a partial order then $\leq$ can always be extended to a linear order. This is sometimes called Szpilrajn´s theorem although it had been previously proved by ...

**1**

vote

**1**answer

250 views

### Self-containing graphs

[Second try, after this question failed.]
Let me sketch a notion of self-containing structures by a simple example. Consider the class $\Gamma$ of finite or countable digraphs ("graphs" for short) ...

**5**

votes

**2**answers

326 views

### Applications of forcing in model theory

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

**4**

votes

**0**answers

191 views

### Schanuel's conjecture and abstract elementary classes

Are there any connections between Schanuel's conjecture and abstract elementary classes. More precisely
Question. Is there any conjecture in abstract elementary classes whose truth implies the ...

**5**

votes

**1**answer

142 views

### Is there a truth approximation on a cumulative hierarchy?

Note to the following well known theorem:
Theorem (1): If $\kappa$ be a "measurable" cardinal and $\mathcal{F}$ be a "non-principal $\kappa$-complete normal" ...

**4**

votes

**1**answer

182 views

### “Small” subfields of algebraically closed fields

Sufficient background:
Let $\mathcal{M}=(M,...)$ be an $\mathcal{L}$-structure and $X\subset M$.
Definition. $X$ is large if there exists a function $f:\mathcal{M}^n \overset {\leq k} \rightarrow ...

**5**

votes

**0**answers

169 views

### Krull dimension and Morley rank

Definition : A Topological space $\mathcal{D}$ is called noetherian if it satisfies the descending chain condition for closed subsets. We define the dimension of $\mathcal{D}$ to be the supremum of ...

**12**

votes

**0**answers

730 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 ...

**17**

votes

**1**answer

734 views

### Main open computational problems in quantifier elimination?

A language is said to have quantifier elimination if every first-order-logic sentence in the language can be shown to be equivalent to a quantifier-free sentence, i.e., a sentence without any ...

**6**

votes

**1**answer

655 views

### Is there a monster behind the trees?

First Fix the following notation:
$\forall \kappa\in Card~~~Tp(\kappa):="\kappa~has~tree~property"$
The large cardinals as "monsters of heaven" live everywhere in the land of ...

**5**

votes

**0**answers

86 views

### Can the isomorphism relation for countable models become harder when adding finitely many constants?

I am particularly interesting in the case where $T$ is o-minimal, but I would be interested in any theory $T$ (or even an $L_{\omega_1,\omega}$-sentence) which has this property.
Context: view the ...

**2**

votes

**0**answers

360 views

### Deducing Skolem's nonstandard integers from downward Lowenheim-Skolem?

If one has a nonstandard model $\mathcal{N}$ of PA and adjoins to the first-order theory the countable list of axioms $1<H,\, 2<H,\, 3<H, \ldots$ (satisfied in $\mathcal{N}$) for all the ...

**6**

votes

**0**answers

172 views

### Axiomatizations of the real exponential field

According to Marker's "Model Theory: An Introduction", the real exponential field has a $\forall\exists$ axiomatization (because it is model complete) but no-one has any idea what such an ...

**3**

votes

**0**answers

85 views

### Is a proconstructible subsemigroup of $M_n(\mathbb{C})$ an intersection of constructible subsemigroups?

Let $S$ be a proconstructible subsemigroup of $M_n(\mathbb{C})$, that is a subsemigroup which is an intersection of constructible sets. Is $S$ an intersection of constructible subsemigroups?
The ...

**6**

votes

**1**answer

229 views

### Vopenka's Principle for non-first-order logics

(For simplicity, the background theory for this post is NBG, a set theory directly treating proper classes which is a conservative extension of ZFC.)
Vopenka's Principle ($VP$) states that, given any ...

**3**

votes

**0**answers

158 views

### strong order property in continuous logic

I am wondering what is the accepted version of the strong order property in continuous logic.
The definition for classical logic is as follows:
$T$ has SOP$_n$ (for $n\geq 3$) if there is a formula ...

**14**

votes

**1**answer

202 views

### Existence property for ordered fields

A theory $T$ has the existence property (EP) if the following holds:
Let $\phi(x)$ be a formula with one free variable (and no parameters) such that $T \vdash (\exists x) \phi(x)$. Then there is ...

**3**

votes

**1**answer

400 views

### Suggestions on the best introductory Model Theory texts

Any recommendations on the best texts for introducing Model Theory?

**1**

vote

**0**answers

60 views

### a regular type of U-rank > 1 in the theory of compact complex spaces

What is an example, in the theory of compact complex spaces, of a regular (i.e. orthogonal to all its forking extensions) type which is of U-rank strictly greater than 1?
update: after witnessing ...

**1**

vote

**2**answers

270 views

### Embedding of consistent subset in first order logic (finitely many variables)

I am looking at FOL with no equality, no constant, no function symbol and the unique binary predicate $\in$ with variables in arbitrary sets $V$ or $W$. Specifically we define ${\bf P}(V)$ as the free ...

**2**

votes

**0**answers

122 views

### topology generated by irreducible componets of $\Gamma$-invariant closed sets

For an analytic space $U$ equipped with an action of a group $\Gamma$,
call a subset $Z\subseteq U$ $\Gamma$-closed iff
it is a closed analytic subset and each of its irreducible components
is an ...

**7**

votes

**1**answer

147 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 ...

**7**

votes

**1**answer

233 views

### Iterating definability

An odd -- probably basic -- question about model theory:
For $\mathcal{M}$ a structure in a (first-order) signature $\Sigma$, let $\mathcal{M}'$ be the structure in signature $\Sigma\sqcup\lbrace ...

**6**

votes

**0**answers

228 views

### Is there Ultracoproduct-like construction for topological spaces in general?

In
http://arxiv.org/pdf/math/9704205.pdf
they define the ultracoproduct of a sequence of compact Hausdorff spaces, $\sum_\mathcal{U}X_i$ along an ultrafilter $\mathcal{U}$ as the Wallman-Frink ...

**3**

votes

**0**answers

147 views

### Saturated Ehrenfeucht-Mostowski models

Inspired by this question on MSE I tried to prove the following:
Let $T$ be a complete theory in a first order language and $\kappa$ a cardinal. If $T$ is $\kappa$-stable, then there exists a ...

**30**

votes

**1**answer

924 views

### Is $π$ definable in $(\Bbb R,0,1,+,×,<,\exp)$?

(This question is originally from Math.SE, where it didn't receive any answers.)
Is there a first-order formula $\phi(x) $ with exactly one free variable $ x $ in the language of ordered fields ...

**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

725 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

190 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 $x+z \prec y + z$ ...