**4**

votes

**2**answers

207 views

### Natural examples of $\bf\Sigma^0_3$ equivalence relations

I have been reading about Borel equivalence relations and I have noticed that while $\bf\Sigma^0_3$ equivalence relations are mentioned, there is a conspicuous absence of natural examples (other than ...

**1**

vote

**1**answer

147 views

### Injective subset function

Let $X$ be a non-empty set and let $F: X \to {\cal P}(X)$ be a function with the following property:
for $A \subseteq X$ we have $|A| \leq |\bigcup F(A)|$.
Does this imply that there is an ...

**3**

votes

**1**answer

244 views

### On generic forcing conditions

Let $P$ be a forcing poset, and $Q \in V^P$ a forcing poset in $V^P$. Let $M \prec H(\lambda)$ ($\lambda$ sufficiently large) countable with $P,Q \in M$.
What I want to know is if then the following ...

**6**

votes

**1**answer

182 views

### Consistency of the nonrigidity of $P(\omega_1)/NS$

Is it consistent with ZFC that there exists an automorphism of $P(\omega_1)/\mathrm{NS}_{\omega_1}$ which is not the identity?

**2**

votes

**0**answers

115 views

### Defining Global Choice in terms of strong limit cardinals over $ZF$

In his answer to user33038's mathoverflow question "What axioms are stronger than the Axiom of choice?", Prof. Hamkins writes:
"What's more, the axiom of choice is equivalent over $ZF$ to the ...

**4**

votes

**1**answer

168 views

### Large Cardinal Principles that Imply $\Sigma_3^1$-Generic Absoluteness

It is known that (light-face) $\Sigma_3^1$ generic absoluteness is consistent with $\mathsf{ZFC}$: Friedman and Bagaria showed that it holds in the $\text{Coll}(\omega, < \kappa)$ extension of $V$ ...

**2**

votes

**0**answers

125 views

### RCS iteration such that the RCS limit is semi-proper

For a countable ordinal $\eta$ and an ordinal $\gamma$ let $\langle P_\alpha, \dot{Q}_\alpha \colon \alpha < \gamma \rangle$ be an RCS iteration with RCS limit $P_\gamma$, such that
...

**2**

votes

**2**answers

199 views

### Hedetniemi's conjecture for graphs with countable chromatic number

Are there graphs $G, H$ such that $\chi(G) = \chi(H) = \aleph_0$, but $\chi(G\times H) < \aleph_0$?

**6**

votes

**2**answers

238 views

### A question regarding strong cardinals and measure sequence

Let $E$ be a $(\kappa, \lambda)$-extender such that $j: V\to M\simeq Ult(V,E)$ is the corresponding elementary embedding with critical point $\kappa$, $M\supset V_{\kappa+2}$, $M^\kappa\subset M$. Let ...

**6**

votes

**0**answers

159 views

### rigidity of $\mathcal P(\omega_1) / NS$ under MA

In Woodin's book, Lemma 5.100 asserts that if $MA_{\omega_1}$ holds and there is an $\omega_2$-saturated ideal $I$ on $\omega_1$, then $\mathcal P(\omega_1)/I$ is a rigid boolean algebra, meaning it ...

**2**

votes

**0**answers

120 views

### What other axioms for set theory can be written in the form: “If mathematical structures $X$ and $Y$ are equipotent, then they're isomorphic”?

The "injective continuum function hypothesis" (ICF) is the following statement.
ICF (Version 0). For all cardinal numbers $\kappa$ and $\nu$, we have $2^\kappa = 2^\nu \rightarrow \kappa = \nu.$
...

**8**

votes

**1**answer

433 views

### Does there exist an uncountable partition of a Polish space so that the union of any collection of blocks is Borel?

Is it consistent that there exists a partition $P$ of the real number line $\mathbb{R}$ such that $|P|>\aleph_{0}$ but where $\bigcup R$ is Borel whenever $R\subseteq P$?
If ...

**4**

votes

**1**answer

295 views

### “set of all irreducible representations of a group”, set-theoretic issues [closed]

I am working on a problem related to representations of the Weil group of a local field $\mathcal{W}_F$. In many articles one introduces the set $\hat{\mathcal{W}}_F$ of all equivalence classes of ...

**1**

vote

**0**answers

216 views

### Some questions regarding Shelah's revised Generalized Continuum Hypothesis [closed]

It is well known that $\mathsf{ZF}+\mathsf{GCH}\vdash\mathsf{AC}$ (which means that the Kunen inconsistency can be proven in $\mathsf{ZF}+\mathsf{GCH}$). Consider now Shelah's revised Generalized ...

**8**

votes

**2**answers

315 views

### Has by well-foundedness every non-empty class an $R$-minimal element? Also if axiom REG is not assumed?

I asked this question here on Math.SE but uptil now it was not answered. So I decided to give it a try. Thank you in advance.
Working in $\mathbf{ZF}$ let $R$ be a proper class of ordered pairs that ...

**14**

votes

**3**answers

449 views

### Does every set $X$ have a topology for which the only continuous self-surjection is the identity map?

This question is a special case of Dominic van der Zypen's question Reconstructing relations with the image relation of a topology, as discussed in the comments, particularly the comment of Eric ...

**3**

votes

**0**answers

145 views

### How distributive are the bad Laver tables?

Suppose that $n\in\omega\setminus\{0\}$. Then define $(S_{n},*)$ to be the algebra where $S_{n}=\{1,...,n\}$ and $*$ is the unique operation on $S_{n}$ where
$n*x=x$
$x*1=x+1\,\text{mod}\, n$ and
if ...

**6**

votes

**1**answer

191 views

### Reverse of a termspace forcing fact

Suppose $\kappa$ is an inaccessible cardinal. Consider the termspace forcing for adding a Cohen subset of $\kappa$ after $\mathbb P= Col(\omega,<\kappa)$. Members are Levy names for countable ...

**11**

votes

**2**answers

382 views

### Connected but no path connected components

Is there a Borel subset of plane which is connected but whose only path connected components are singletons?
I know that a Bernstein set is a non Borel example of such a set. Thanks!

**0**

votes

**1**answer

116 views

### Cardinality of pairwise non-isomorphic complete lattices on an infinite cardinal $\kappa$

Suppose $\kappa$ is an infinite cardinal. Let $\cal S$ be a collection of pairwise non-isomorphic complete lattices on the ground set $\kappa$. What cardinality can $\cal S$ have at most? Is the ...

**13**

votes

**1**answer

629 views

### Does “cardinal arithmetic is well-defined” imply axiom of choice?

Let me quickly explain what I mean with my question.
Let $(\kappa_i)_{i\in I}$ be a collection of cardinal numbers, indexed by elements of some set $I$. We can try to define $\sum\limits_{i\in ...

**7**

votes

**1**answer

187 views

### Consistency strength of being strong cardinal and indestructible under collapses

What is the consistency strength of the following statement:
$\kappa$ is a strong cardinals and it is indestructible under $Col(\kappa, <\theta),$ where $\theta> \kappa$ is some fixed ...

**11**

votes

**1**answer

257 views

### Does stationary reflection imply Mahloness?

Suppose $\kappa$ is strongly inaccessible and every stationary subset of $\kappa$ reflects. Must $\kappa$ be Mahlo?
Remarks:
It is possible for every stationary subset of $\kappa$ to reflect, but ...

**4**

votes

**1**answer

175 views

### Can we always add sets without collapsing cardinals or adding [very] bounded sets?

Given a model of $\sf ZFC$, and an infinite ordinal $\alpha$. Can we prove that there is always a cardinal $\kappa$, and a forcing $\Bbb P$, such that:
$\Bbb P$ does not add sets of rank ...

**6**

votes

**2**answers

416 views

### Collapsing the cardinals between two singular cardinals

Question 1: Is it consistent that there is a forcing notion collapsing $\aleph_{\omega\cdot 2}$ to $\aleph_\omega$ without collapsing $\aleph_\omega$ or $\aleph_{\omega\cdot 2 + 1}$?
If the ...

**-1**

votes

**1**answer

244 views

### Class forcings and elementary embeddings

In the Hamkins-Kirmayer-Perlmutter paper "Generalizations Of The Kunen Inconsistency", they prove the following theorem:
"Theorem 7: In any set forcing extension $V[G]$, there is no nontrivial ...

**5**

votes

**1**answer

254 views

### Transcendental distance sets

Define a set $S \subset \mathbb{R}^d$ as a
transcendental distance set if the distance between any pair of
distinct points of $S$ is transcendental.
For example, $S = \{ k \, \pi \;\mid\; k=1,2,\ldots ...

**6**

votes

**0**answers

180 views

### $\alpha$-minimal degrees for singular $\alpha$

An important question in $\alpha$-recursion theory is whether there is a minimal $\alpha$-degree at $\alpha=\aleph_\omega.$
Question 1. Who first introduced the above question, and where can I find ...

**14**

votes

**2**answers

698 views

### A question of Erdős

In the following paper (pages 122-23), Erdős asks if there is a constant $c > 0$ such that every subset $A$ of plane of area more than $c$ contains the vertices of a triangle of unit area.
Is this ...

**9**

votes

**1**answer

583 views

### Are there discontinuities in the large cardinal hierarchy?

Suppose that $\Phi(x,y)$ is a formula in the language of set theory so that for each natural number $n$, the axiom $\exists x\Phi(n,x)$ is a large cardinal axiom (for example consider $n$-huge ...

**15**

votes

**0**answers

395 views

### What sort of cardinal number is the Löwenheim-Skolem number for second-order logic?

In their paper "On Löwenheim-Skolem-Tarski numbers for extensions of first order logic", Magidor and Väänänen make the following statement:
"For second order logic, $LS(L^{2})$ ...

**7**

votes

**1**answer

312 views

### A question related to Woodin's $HOD$ conjecture

Recall that an $\kappa$ is $\omega$-strongly measurable in $\text{HOD}$ if there exists
$\lambda < \kappa$ such that $(2^{\lambda})^{\text{HOD}} < \kappa$ and such that
there is no partition of ...

**10**

votes

**2**answers

386 views

### Constructing an $\omega_1$-sequence of functions that almost extend all previous functions

I want to construct a sequence of functions $$f_\alpha: \alpha \rightarrow \omega,\ \alpha < \omega_1$$
such that for all $\alpha < \omega_1$ the following holds:
$f_\alpha$ is injective.
...

**10**

votes

**1**answer

241 views

### Posets preserving stationary subsets of $\omega_1$ and no new $\text{cof}(\omega)$ ordinals, but without countable covering property

What are some examples of posets $\mathbb{P}$ which have the following properties? It's OK if the definition uses large cardinals or some other hypothesis.
$\mathbb{P}$ preserves stationary subsets ...

**29**

votes

**2**answers

1k views

### Measurability and Axiom of choice

In some situations, you need to show Lebesgue-measurability of some function on $\mathbb{R}^n$ and the verification is kind of lengthy and annoying, and even more so because measurability is "obvious" ...

**0**

votes

**1**answer

202 views

### A question regarding extendible cardinals and a result of M. Magidor

The following definitions and Theorems come from M. Magidor's paper "On the Role of Supercompact and Extendible Cardinals in Logic" (Israel J. Math., Vol. 10, 1971):
"Definition: Logic is called ...

**7**

votes

**1**answer

598 views

### Is every real a member of some CTM?

I read somewhere about the hyperuniverse of countable transitive models of ZFC (http://www1.maths.leeds.ac.uk/maloa/lecturenotes/RW3%20Munster/friedman.pdf). It states an assumption, namely that every ...

**7**

votes

**1**answer

433 views

### Why is the set-theoretic principle $\diamondsuit$ called $\diamondsuit$?

A shallow answer would just point to theorem 6.2 in Jensen's 1972 paper "The fine structure of the constructible hierarchy", where Jensen introduces this property. Or was this symbol used already ...

**4**

votes

**1**answer

267 views

### Plausibility argument for a measurable cardinal

The following question is not mathematically precise but perhaps of some philosophical interest.
A typical plausibility argument for assuming the existence of inaccessible cardinals goes as follows: ...

**5**

votes

**1**answer

186 views

### References for Forcing with Side Conditions

I'm looking for some good references about Forcing with Side Conditions, including expository papers that explain the main ideas with some details in order to give me a fairly clear insight of those ...

**5**

votes

**1**answer

271 views

### Diagonalizing against a non stationary set of functions

Suppose $\kappa$ is regular uncountable (assume $2^{< \kappa} = \kappa$ and $\kappa$ is weakly Mahlo if needed). Is the following true?
For every sequence $\langle f_i: i \to 2 \mid i \in A ...

**1**

vote

**1**answer

205 views

### Well-ordered reference

I'm wondering if there is a standard reference for the following straightforward facts about well-orderings. (They are quite easy to prove, I just need a handy/standard reference.)
Let $(S,<)$ ...

**10**

votes

**2**answers

666 views

### A question of Erdos on entire functions

At the end of the following paper, Erdos asked if there is a family $F$ of entire functions of size continuum such that for every $z \in \mathbb{C}$, $\{f(z) : f \in F\}$ has size less than continuum. ...

**5**

votes

**1**answer

156 views

### Does the critical sequence for subalgebras of elementary embeddings with finitely many generators have order type $\omega$?

Suppose that $\lambda$ is a cardinal. Let $\mathcal{E}_{\lambda}$ be the set of all elementary embeddings from $V_{\lambda}$ to $V_{\lambda}$. If $j,k\in\mathcal{E}_{\lambda}$, then define
...

**-1**

votes

**2**answers

324 views

### Subsets of $\mathbb{N}$ whose lower density respects complements

The lower density of $A\subseteq\mathbb{N}$ is defined to be $\lambda(A)=\lim\text{inf}_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}$. We set $${\cal C} = \{A\subseteq \mathbb{N}: ...

**8**

votes

**0**answers

265 views

### preserving saturated ideals

A reliable source made the following claim:
Suppose CH there is an $\omega_2$-saturated ideal on $\omega_1$. Then this is preserved by $\mathrm{Add}(\omega_1,\omega_2)$.
Question 1: How do you ...

**2**

votes

**1**answer

163 views

### Implications between different covering properties of spaces

Let $X$ be a set. A set ${\cal C}\subseteq {\cal P}(X)$ is said to be a cover of $X$ if $\bigcup {\cal C} = X$ and $X\notin {\cal C}$.
If ${\frak U}$ and $\frak{W}$ are collections of covers of a ...

**5**

votes

**0**answers

159 views

### Sierpinski sets and extensions of Lebesgue measure

I am duplicating an old problem from stackexchange:
Suppose that every countably generated sigma algebra extending the Borel sigma algebra on $[0, 1]$ admits a measure extending the Lebesgue measure ...

**1**

vote

**1**answer

146 views

### $\mathbb{P}_{\kappa}$ forces non$(\mathcal{M})$=cov$(\mathcal{M})=\kappa$

Let $\kappa$ be an uncountable regular cardibnal. Consider the finite support iteration $(\langle \mathbb{P}_{\alpha}\rangle _{\alpha \leq \kappa},\langle \mathbb{\dot{Q}}_{\alpha}\rangle _{\alpha ...

**9**

votes

**2**answers

513 views

### Preserving $\omega_1$ is Inaccessible to the reals

$\omega_1$ is inaccessible to the reals if and only if for all $x \in {}^\omega\omega$, $\omega_1^{L[x]} < \omega_1$.
The question is if $\omega_1$ is inaccessible to the reals in $V$ and ...