**1**

vote

**1**answer

285 views

### $(\kappa , \lambda)$ - Minimal Models of $\text{ZF}$

The notion of minimality in model theory is related to the existence of a gap in the size of definable subsets of a model. Now consider the following generalization:
Definition 1: Let $M$ be a ...

**3**

votes

**1**answer

854 views

### From elementary equivalence to isomorphism

A few years ago, when I took the basic course in Logic, I was very surprised to discover that given a signature $\sigma$ and two structures $M$ and $N$ of $\sigma$ which are elementarily equivalent ...

**3**

votes

**2**answers

160 views

### Joint Forcing Extension Property

Definition 1: A class $\mathcal{K}$ of countable transitive models of $\text{ZF}$ has strong "joint forcing extension property" (JFEP) iff for all $M,N\in \mathcal{K}$ there are forcing notions ...

**2**

votes

**1**answer

135 views

### Expressing “There are more Fs then Gs” in first-order logic

Is there a sentence of first-order logic that's true in all finite interpretations in which there are more Fs than Gs, and false in all finite interpretations in which there are not more Fs than Gs?
...

**6**

votes

**0**answers

105 views

### What classes of complex manifolds are known to be definable in an o-minimal expansion of the real field?

It is a widely known (perhaps slightly folkloric) fact that compact complex manifolds, understood as first-order structures with a predicate for each analytic subset, are definable in an expansion of ...

**3**

votes

**0**answers

120 views

### Cardinality of Grothendieck ring in model theory

Good evening,
In model theory there is a notion of Grothendieck ring defined here http://math.berkeley.edu/~scanlon/papers/greu12jun00.pdf.
Do we know anything about the cardinality of these rings ?
...

**9**

votes

**1**answer

667 views

### Shelah's book on “Classification Theory”

As we know one of the most important and fundamental books in stability, simplicity, forking and ... classification theory, is Shelah's "Classification Theory" where lots of original ideas of the ...

**3**

votes

**3**answers

296 views

### Turing Functional and $\Sigma_1^0$-formulas in models of fragments of PA

In models of PA with restricted induction power (for example, only $I\Sigma_n$ is present), the failure of higher induction scheme is characterised by the existence of definable cuts (like $\Sigma_2$ ...

**12**

votes

**3**answers

710 views

### The interplay between certain aspects of interpretability, model theory and category theory

I have some questions about the interplay of interpretability, model theory and category theory. Since I had difficulties in finding literature or other helpful information about this topic, it would ...

**7**

votes

**1**answer

299 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 ...

**7**

votes

**1**answer

235 views

### Is the consistency of $\mathcal{L}_{\infty\omega}$-sentences absolute?

The question is exactly that of the title. Suppose $\varphi\in V$ is an $\mathcal{L}_{\infty\omega}$-sentence, and $W$ is an inner model of $V$ such that $\varphi\in W$. Is the statement
...

**5**

votes

**0**answers

383 views

### Recent application of model theory in set theory by Shelah-Malliaris

Recently one of the oldest open problems in set theory about the cardinal invariants of the continuum (i.e the question of whether $\mathfrak{p}=\mathfrak{t}$) was solved by Shelah and Malliaris (see ...

**7**

votes

**0**answers

209 views

### Is there a Rado category?

The Rado graph appears to have a nice universality property (it contains all finite and all countably infinite graphs as induced subgraphs) and homogeinety property (any isomorphism between ...

**4**

votes

**1**answer

171 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

191 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

545 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

398 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 ...

**3**

votes

**2**answers

343 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

184 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

202 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

445 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

851 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 ...

**6**

votes

**1**answer

188 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

200 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 ...

**10**

votes

**1**answer

492 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

313 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

187 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

239 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 ...

**26**

votes

**4**answers

2k 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

446 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 ...

**27**

votes

**3**answers

699 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

208 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 ...

**5**

votes

**1**answer

337 views

### Omitting types and Baire category

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

**10**

votes

**2**answers

383 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 ...

**2**

votes

**1**answer

263 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) ...

**9**

votes

**6**answers

679 views

### Applications of forcing in model theory

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

**5**

votes

**0**answers

207 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

156 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

207 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 ...

**10**

votes

**0**answers

210 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 ...

**17**

votes

**0**answers

843 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 ...

**19**

votes

**1**answer

839 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

689 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 ...

**6**

votes

**0**answers

98 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 ...

**3**

votes

**0**answers

409 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 ...

**17**

votes

**1**answer

402 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

89 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 ...

**9**

votes

**2**answers

373 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 ...

**4**

votes

**0**answers

188 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

215 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 ...