**13**

votes

**0**answers

212 views

### How hard is it to destroy a diamond? (with a real)

If we start with $V\models\lozenge$, it is not hard to force the failure of diamond. You can blow up the continuum, or destroy all the Suslin trees. You can blow up the continuum of $\aleph_1$, and ...

**-3**

votes

**0**answers

51 views

### How to prove that $h^*(A)$ exists and $h^*(A)=\{x\in ON : x \leq^* A\}$? [on hold]

Definition : $h^*(A)$ is the least aleph such that $\not\leq^* A$.
$Z \not\leq^* X$ means theie is no map from $X$ onto $Z$.
How to prove that $h^*(A)$ exists and $h^*(A)=\{x\in ON : x \leq^* ...

**5**

votes

**0**answers

88 views

### Analytic equivalence relations whose classes are sometimes Borel

There are analytic equivalence relations for which the statement "All classes are Borel" is independent of $ZFC$. In all the examples I know about, the classes are non Borel in $L$ or $L[z]$ for some ...

**2**

votes

**1**answer

162 views

### Can the epsilon induction condition be presented with successor case and limit case, similar to transfinite induction?

The epsilon induction looks like this:
$\forall x \Big(\forall y (y \in x \rightarrow P(y)) \rightarrow P(x)\Big) \rightarrow \forall x P(x)$
Here, the quantifiers "run over" any sets and not only ...

**0**

votes

**0**answers

21 views

### Lists of sets as objects of ZF axiomatics [migrated]

I have a naive question about foundations of mathematics. A common opinion of most mathematicians is that the essential part of mathematics can be reduced to ZF(C) axioms. I do not quite understand ...

**-7**

votes

**0**answers

86 views

### what are the practical applications of sets in our daily life? [on hold]

I don`t know the answer to this question?I know I sound stupid writing something in my own question but the computer was forcing me to write something.

**9**

votes

**1**answer

166 views

### Does $Add(\kappa,1)^L$ ever collapse cardinals?

In general, we know that adding a subset to a regular cardinal $\kappa$ can collapse cardinals. If, for example, there is $\gamma < \kappa$ with $2^\gamma >\kappa$, then $Add(\kappa,1)$ will ...

**7**

votes

**3**answers

237 views

### Borel coloring of a graph on the set of all functions $f:\mathbb{N}\to\mathbb{N}$

The following question was asked in a comment by Joel David Hamkins in Graph on the set of all functions $f:\mathbb{N}\to\mathbb{N}$.
Let $V$ be the set of all functions $f:\mathbb{N}\to\mathbb{N}$. ...

**2**

votes

**1**answer

217 views

### Graph on the set of all functions $f:\mathbb{N}\to\mathbb{N}$

Let $V$ be the set of all functions $f:\mathbb{N}\to\mathbb{N}$. Let two distinct functions $f,g:\mathbb{N}\to\mathbb{N}$ form an edge if and only if they differ in exactly one input $n\in\mathbb{N}$.
...

**11**

votes

**2**answers

283 views

### Graph $G$ with $\omega(G) = 2$ but $\chi(G) \geq \aleph_0$

Given an infinite cardinal $\kappa$, is there a graph $G$ that has no clique consisting of more than 2 points, but $\chi(G) = \kappa$?

**3**

votes

**1**answer

310 views

### Is this version of van der Waerden's Theorem consistent with ZFC?

One way to phrase van der Waerden's Theorem is:
For every finite coloring of $\mathbb N$ and every finite $F \subseteq \mathbb N$, there exist $a,b \in \mathbb N$ such that $a + b \cdot F$ is ...

**10**

votes

**2**answers

504 views

### Does van der Waerden's Theorem hold for $\omega_1$?

One way to phrase van der Waerden's Theorem is:
For every finite coloring of $\mathbb N$ and every finite $F \subseteq \mathbb N$, there exist $a,b \in \mathbb N$ such that $a + b \cdot F$ is ...

**4**

votes

**1**answer

144 views

### Order dimension of $\omega^\omega/(fin)$

Let $\omega^\omega$ denote the collection of all functions $f:\omega\to\omega$. For $f,g\in\omega$ we say $f\simeq g$ if and only if $\exists N \in \omega$ such that $f(n) = g(n)$ for all $n\geq N$.
...

**3**

votes

**2**answers

122 views

### ${\frak b}$ and ${\frak d}$ defined with $\leq$ instead of $\leq^*$

Let $\omega^\omega$ denote the collection of all functions $f:\omega\to\omega$. For $f,g\in\omega$ we define
$f\leq g$ if $f(n)\leq g(n)$ for all $n\in\omega$;
$f\leq^* g$ if there is $N\in\omega$ ...

**6**

votes

**1**answer

129 views

### completions of regular suborders

Suppose $\mathbb{P}$ is a regular suborder of the separative partial order $\mathbb{Q}$ (see below for definitions). Must there always exist some complete boolean algebra $\mathbb{B}$ such that:
...

**6**

votes

**0**answers

155 views

### Where does this strengthening of I1 stand?

Let's call a cardinal $\delta$ an $\text{I1}$-tower cardinal if for each $A\subseteq V_{\delta}$, there exists a $\kappa<\delta$ such that whenever $\kappa<\alpha<\delta$ there is some ...

**28**

votes

**0**answers

675 views

### Bidi: A new cardinal characteristic of the continuum?

This question assumes familiarity with combinatorial cardinal
characteristics of the continuum.
Identify an infinite set $a\subseteq\mathbb{N}$ with its increasing
enumeration. Thus, for each natural ...

**4**

votes

**0**answers

109 views

### The dominating number $\mathfrak{d}$ and convergent sequences

All spaces considered below are compact Hausdorff.
If $K$ is a space, then $w(K)$ is its weight. For a Boolean algebra $\mathcal{A}$, $K_\mathcal{A}$ denotes its Stone space. I am interested in ...

**7**

votes

**1**answer

265 views

### Under $\neg CH$, have countable unions of rationally independent numbers inner measure zero?

In their 1943 paper On non-denumerable graphs, Erdos and Kakutani suggest as likely the following proposition.
(EK*) Suppose CH fails and $\lbrace M_n : n \in \omega \rbrace$ is a countable family of ...

**3**

votes

**0**answers

80 views

### Partition refinement of a clopen covering in $\Box (\omega+1)^\omega$

Consider $\omega+1$ with the interval topology, that is $U\subseteq (\omega+1)$ is open if and only if $U\subseteq\omega$ or $(\omega+1)\setminus U$ is finite.
We write $(\omega+1)^\omega$ for the ...

**2**

votes

**1**answer

319 views

### Formulating Kunen's inconsistency and Reinhardt cardinals in term of category theory

It is known that one can formulate certain large cardinal axioms (e.g. Vopenka's principle--see Mike Shulman's answer to Harry Gindi's mathoverflow question "Reasons to believe Vopenka's ...

**1**

vote

**1**answer

264 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

169 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

104 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 ...

**4**

votes

**1**answer

193 views

### What consistency results follow the assumption: $\forall\alpha(\aleph_{\alpha+1}\nleq2^{\aleph_\alpha})$?

In a recent question on Math.SE it was asked whether or not For every infinite cardinal $\mathfrak m$ there is no $\aleph$ number, $\kappa$, such that $\mathfrak m<\kappa<2^{\mathfrak m}$.
By ...

**5**

votes

**1**answer

188 views

### $\text{cov}(\mathcal{M})$ vs. $\mathfrak{b}$ vs. $\mathfrak{s}$

Let me first recall some pretty standard notations:
$\text{cov}(\mathcal{M})$ is the covering number of the ideal $\mathcal{M}$ of all meager subsets of $\mathbb{R}$;
$\mathfrak{b}$ is the bounding ...

**15**

votes

**1**answer

425 views

### Whatever happened to $L(j)$?

So this question probably shows my inner model theoretic ignorance, but:
In "Two remarks on elementary embeddings of the universe" (http://projecteuclid.org/download/pdf_1/euclid.pjm/1102969567), ...

**0**

votes

**0**answers

39 views

### Uncountability of the set of subsets of $\mathbb N$ [migrated]

Denote by $\mathbb{N}$ the set of natural numbers and by $2^{\mathbb N}$ the set of all subsets of $\mathbb N$. Let $E$ be some subset of $2^{\mathbb N}$ such that for every pair of elements in $E$, ...

**9**

votes

**1**answer

276 views

### Assuming AD, is every infinite cardinal closed under power set in a choice model?

Assume AD+DC. Assume $\kappa$ is an infinite cardinal and $N$ is a (set or class) transitive model of ZFC containing $\kappa$.
Is it true that for all $\alpha<\kappa$, $N$ thinks that the power ...

**7**

votes

**0**answers

184 views

### A question about finitely additive extensions of Lebesgue measure

Suppose $m:P([0, 1]) \to [0, 1]$ is a finitely additive measure extending the Lebesgue measure. Must there exist some $X \subseteq [0, 1]$ such that $m(X \cap I) = |I|/2$ for every sub interval $I ...

**5**

votes

**1**answer

165 views

### Universally Baire Tree Representation of Projective Sets

In Feng, Magidor, and Woodin "Universally Baire Sets of Reals", they show that if $A$ is a $\mathbf{\Pi}_2^1$ set and $U$ and $V$ are any pair of trees witnessing the universal baireness of $A$, then ...

**5**

votes

**2**answers

162 views

### Extending hyperconnected spaces

A hyperconnected space is a topological space such that every two non-empty open sets have non-empty intersection. Let's call a space $(X,\cal{T})$ maximally hyperconnected if it is hyperconnected and ...

**1**

vote

**0**answers

122 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 ...

**3**

votes

**1**answer

224 views

### A question about Cantor's Power Set theorem without the Axiom of Choice

Assume that we are working in ZF set theory without the Axiom of Choice. If S is an infinite set, let $S(f)$ denote the set of all finite subsets of $S$, let $S(I)$ denote the set of all infinite ...

**5**

votes

**0**answers

143 views

### Reference to forcing with a sigma ideal $\cong$ Cohen forcing

This is a historical question: Who was the first person to notice the following?
If $V \models \kappa$ is measurable and $P$ adds $\kappa$ Cohen reals, then in $V^P$, letting $\hat{I}$ to be the ...

**-2**

votes

**0**answers

10 views

### Partition on a Closed Set A= [2,3] [migrated]

Is it possible to define a partition on a closed set,such that the union of the partitions will give [2,3] and their intersection to be empty?

**9**

votes

**1**answer

165 views

### Are the failure of SCH and “$cf([\mu]^{cf (\mu)},\subset)>\mu^+$ for some singular” equiconsistent?

Is it true that the following two statements are equiconsistent?
(1) $2^\mu>\mu^+$ for some strong limit singular cardinal $\mu$
(2) $cf([\mu]^{cf (\mu)},\subset)>\mu^+$ for some singular ...

**4**

votes

**1**answer

174 views

### Meager set of full measure

Let X be a compact Hausdorff topological group and let m be the Haar measure on X. Can we find a meager set in X whose complement is m-null? I can do it when X is separable but I don't know if there ...

**0**

votes

**0**answers

33 views

### How can nontrivial elementary embeddings of the universe to some inner model be surjective? [migrated]

Consider $\kappa$ to the least measurable cardinal, or equivalently $\kappa$ is the critical point for an elementary embedding $j:V \rightarrow M$ from the universe $V$ to an inner model $M$ (critical ...

**3**

votes

**0**answers

160 views

### name for an intermediate notion between huge and 2-huge

I am employing a large cardinal notion that has been used explicitly before, and I am wondering if someone has given it a good succinct name.
A cardinal $\kappa$ is huge if there is an elementary $j ...

**6**

votes

**0**answers

244 views

### Reference for “if the set $A$ is Suslin, then every $\Sigma^1_1(A)$ set is Suslin”

Does anyone know of a reference for one or both of the following facts (in $\mathsf{ZF}$)?
If the set of reals $A$ is Suslin, then every $\Sigma^1_1(A)$ set of reals is Suslin.
If $T$ is a tree on ...

**13**

votes

**2**answers

877 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 ...

**7**

votes

**1**answer

355 views

### Explicit counter example to Vopěnka's principle in the constructible universe?

Vopěnka's principle is a large cardinal axiom which has many equivalent formulations. One of them, which I find especially appealing, is the following: if the universe is satisfies Vopěnka's principle ...

**2**

votes

**1**answer

174 views

### Uncountable cardinals and Prufer $p$-groups

Let $A$ be an elementary Abelian uncountable $p$-group. Is it known if there is an action of a Prufer $q$-group (here $q$ is a prime not necessarily distinct from $p$) $C_{q^{\infty}}$ onto $A$ such ...

**3**

votes

**0**answers

120 views

### A Result of Anders Bjorner: Matchings in countably infinite geometric lattices of finite height

Let $L$ be a countably infinite geometric lattice of finite height $r\ge3$. (A geometric lattice of height $r$ is an atomistic semimodular lattice such that every maximal chain has $r+1$ elements.)
...

**1**

vote

**0**answers

75 views

### Outer measure preserving bijection

Suppose X is a Sierpinski set (So X is uncountable and every null subset of X is countable). Let f be a bijection on X. Must/Does there exist a non null subset Y of X such that for every subset W of ...

**1**

vote

**0**answers

170 views

### The patterns of possibility for nontrivial automorphisms and nontrivial elementary embeddings of the universe

In their paper "The Role of the Foundation Axiom in the Kunen Inconsistency" (arXiv:1311.0814 [Math.LO]), Daghighi, Golshani, Hamkins, and Jerabek show that the patterns of possibility for the ...

**10**

votes

**3**answers

483 views

### Axiom of choice for sets of finite sets

The question I am going to ask is really to satisfy my curiosity, as I am not at all an expert of the subject and do not plan to really work on it. Hence, if you think the question is not suitable for ...

**7**

votes

**1**answer

172 views

### Characterizing L(R) Cardinals in HOD

We're working in L(R) under AD.
We know that
$\omega_1$ is the least measurable in HOD, $\Theta$ is the least woodin, $\delta^2_1$ is the least strong to the woodin, etc.
My question is about ...

**7**

votes

**1**answer

306 views

### Constructing unnatural transformations

In a nutshell, the question is: is it true that any explicit (not involving axiom of choice) pointwise transformation between sufficiently complicated functors is natural almost everywhere?
Let $C$ ...