forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set theory, independence, axiom of choice, continuum hypothesis, determinacy, Borel equivalence relations,...

**-1**

votes

**0**answers

29 views

### Sørensen-Dice coefficient equivalence [on hold]

I'm working on a open source custom NLP engine.
Trying to figure out some similarity measures, I wrote this formula (SIM) used to compare the similarity of two sets:
$SIM = 1 - \frac{|A-B|+|B-A|}{|A|+...

**0**

votes

**1**answer

86 views

### Support of a regular measure Reg

Let $K$ be compact, Hausdorff space but not necessarily metrizable. Let $\mathfrak{M}$ be the Borel $\sigma$ field over $K$ and $\mu$ be a positive Regular Borel measure on $K$. Let $S$ be a subset of ...

**-5**

votes

**0**answers

31 views

### How to prove Quantified statements with different variables [on hold]

I would like to know how could i prove this quantified statment knowing that the universal set={a,b}
Prove that ?x?yP(x,y) impiles ?y?xP(x,y) is always true (=>)

**2**

votes

**1**answer

141 views

### Is $\in$-induction provable in first order Zermelo set theory?

Are there models of first order Zermelo set theory (axiomatized by: Extensionaity, Foundation, empty set, pairing, set union, power, Separation, infinity) in which $\in$-induction fail?
I asked this ...

**3**

votes

**0**answers

104 views

### Ordinal analysis and nonrecursive ordinals

Ordinal analysis is typically described as characterizing recursive ordinals in a theory $T$, but there is a sense in which it can characterize all $T$-ordinals, even those that are nonrecursive.
...

**5**

votes

**1**answer

270 views

### Can a Shelah semigroup be commutative?

A semigroup $S$ is called
$\bullet$ $n$-Shelah for a positive integer $n$ if $S=A^n$ for any subset $A\subset S$ of cardinality $|A|=|S|$;
$\bullet$ Shelah if $S$ is $n$-Shelah for some $n\in\...

**10**

votes

**1**answer

375 views

### Completeness number of ultrafilters

In what I write below, by "ultrafilter" I mean a non-principal ultrafilter.
Given an ultrafilter $U$ on some set $S$, let $\mu$ be the least cardinal such that $U$ is $\mu$-complete but not $\mu^+$-...

**5**

votes

**1**answer

226 views

### $\Delta^{1}_{2}$ and degrees of constructibility $\textbf{on sets}$

This is a follow-up to my question on $\Delta^{1}_{2}$ and degrees of constructibility of real numbers that was answered by the user "William", see here: Can $\Delta^{1}_{2}$ separate degrees of ...

**6**

votes

**0**answers

138 views

### On the number $n_0$ in Shelah's construction of a Jonsson group

In the paper "On a Problem of Kurosh, Jonsson Groups, and Applications", Shelah proved the following
Theorem 2.1. There exists a number $n_0$ such that for every infinite cardinal $\lambda$ with $\...

**-4**

votes

**0**answers

404 views

### Turning a category into a semigroup

Consider a small (that is objects and morphisms belonging to a Grothendieck universe) category $\mathcal{C}$, whose objects are all small sets, and an identity-on-objects functor ${\uparrow}: \mathbf{...

**3**

votes

**0**answers

144 views

### On Khelif's example of a group of countable cofinality having the Bergman property

A group $G$ is defined to have the Bergman property if for any subset $X=X^{-1}$ generating $G$ there exists $n$ such that $X^n=G$.
By a result of Bergman, the permutation group of any set has the ...

**12**

votes

**1**answer

492 views

### A Shelah group in ZFC?

In his famous paper "On a problem of Kurosh, Jonsson groups, and applications" of 1980, Shelah constructed a CH-example of an uncountable group $G$ equal to $A^{6641}$ for any uncountable subset $A\...

**3**

votes

**0**answers

100 views

### Completely I-non-measurable unions in Polish spaces

Problem. Let $X$ be a Polish space, $\mathcal I$ be a $\sigma$-ideal with Borel base, and $\mathcal A\subset\mathcal I$ be a point-finite cover of $X$. Is it true that $\mathcal A$ conatins a ...

**10**

votes

**1**answer

748 views

### Relation between the Axiom of Choice and a the existence of a hyperplane not containing a vector

In a lot of problems in linear algebra one uses the existence, for each $E$ vector space over a field $k$, and each $x\in E$, of a Hyperplane $H$ such that $E=k\cdot x \oplus H$ (Let us denote $\...

**0**

votes

**0**answers

218 views

### Is there a known shorter axiomatization of NF than this?

Is there an already known axiomtization of $NF$ that is shorter than the following axiomatic system in first order logic with equality $``="$ and membership $``\in"$? And what is exactly meant by ...