Questions about the continuum hypothesis.

**11**

votes

**1**answer

200 views

### The Parovichenko cardinal, is it equal to $\max\{\aleph_2,\mathfrak p\}$?

Let us define the Parovichenko cardinal $\mathfrak{P}$ as the largest cardinal $\kappa$ such that each compact Hausdorff space $K$ of weight $w(K)<\kappa$ is the continuous image of the remainder $...

**21**

votes

**0**answers

346 views

### Is it still an open problem whether $\mathbb{R}^\omega$ is normal in the box topology?

On page 205 of his Topology book, James Munkres makes an interesting remark:
It is not known whether $\mathbb{R}^\omega$ is normal in the box topology. Mary-Ellen Rudin has shown that the answer ...

**7**

votes

**1**answer

290 views

### A new cardinal characteristic of the continuum?

Let $\kappa$ be the smallest cardinality of a family $\mathcal F$ of subsets of $\omega$ such that for any bijective function $f:A\to B$ between disjoint infinite subsets of $\omega$ there exists a ...

**19**

votes

**2**answers

1k views

### Mathematical Evidence Backing $|\mathbb{R}|=\aleph_2$

The "true" size of the real line, $\mathbb{R}$, has been the subject of Hilbert's first problem. Due to the Goedel and Cohen's work on the inner and outer models of $\text{ZFC}$, it turned out to be ...

**3**

votes

**1**answer

305 views

### Strong Total Failures vs. Weak Instances of the Generalized Continuum Hypothesis

The exponentiation operator inflicts a subtle information loss on the transfinite numerical equations, pretty similar to the case of $a^2=b^2 \nRightarrow a=b$ in real numbers. In fact, for the ...

**28**

votes

**2**answers

2k views

### On the probability of the truth of the continuum hypothesis

First note that there exists a natural measure $\mu$ on $P(\omega \times \omega)$, inherited from the Lebesgue measure on the reals (by identifying the reals with $P(\omega)$ and $\omega$ with $\omega ...

**2**

votes

**3**answers

202 views

### Example of an $\omega_1$ decreasing chain of dense semicontinua?

In his well-known paper Bellamy constructs an indecomposable continua with exactly two composants. The setup is as follows:
We have an inverse-system $\{X(\alpha); f^\alpha_\beta: \beta,\alpha < \...

**19**

votes

**1**answer

670 views

### On Erdos-Kakutani like Equivalents of the Failure of Continuum Hypothesis

Among all mysterious equivalents of the Continuum Hypothesis and its negation, there is an algebraic combinatorial equivalent of the $\neg CH$ by Erdos and Kakutani (MR0008136) as follows:
...

**13**

votes

**1**answer

375 views

### Continuum Hypothesis and the fact that every co-finite topological space, with uncountable underlying set , is contractible

Let $X$ be a co-finite topological space. If $|X| \ge 2^{\aleph_0}=\mathfrak c$, then $X$ is contractible (https://en.wikipedia.org/wiki/Contractible_space) . Indeed, there is a bijection $f: X \times ...

**7**

votes

**0**answers

241 views

### A strange planar set and the Continuum Hypothesis

Call a number abnormal if its decimal expansion doesn't feature every digit an infinite number of times. Call a triangle in ${\Bbb R}^2$ abnormal if at least one of its angles spans an abnormal ...

**8**

votes

**1**answer

311 views

### Is $[0,1]$ a disjoint union of $\aleph_1$ compact subsets with empty interior?

Is $[0,1]$ a disjoint union of $\aleph_1$ compact subsets with empty interior?
The answer is obviously yes assuming the continuum hypothesis. Also, by Baire's lemma, the answer is negative if one ...

**8**

votes

**2**answers

555 views

### Is it possible to define cardinals that are distinct from either the $\aleph$ numbers or $\beth$ numbers?

I am wondering if there are ways of defining "structure" on infinite sets that generate sequences of cardinals that cannot be proved to have the same cardinality as either the $\aleph$ or $\beth$ ...

**18**

votes

**2**answers

774 views

### Does $V = \textit{Ultimate }L$ imply GCH?

In his Midrasha Mathematicae lectures ("In Search of Ultimate $L$", BSL 23 [2017]: 1–109), Woodin notes that $V = \textit{Ultimate }L$ implies $\textrm{CH}$ (Theorem 7.26, p.103). Is it known whether $...

**18**

votes

**0**answers

559 views

### Is this Variation of the Continuum Hypothesis Inconsistent with ZFC or ZF?

It is a well-known fact that the Generalized Continuum Hypothesis is undecidable from ZFC. For similar sentences $\phi$, this is simply equivalent to ZFC having a model $M$ for which $M\models\phi$.
...

**7**

votes

**1**answer

546 views

### Is it known whether or not $\aleph_\alpha=\beth_\alpha$ can be proven by ZFC?

Given a cardinal number $\aleph_\alpha$, is it known whether or not $\aleph_\alpha=\beth_\alpha$ is independent of ZFC?
One could define $\mathrm{CH}(\aleph_\alpha)$ as $\aleph_\alpha=\beth_\alpha$. ...

**2**

votes

**0**answers

300 views

### In Cohen's Independence of Continuum Hypothesis, can someone explain to me his definition of the $a_\delta$'s at the beginning of the paper?

Cohen's paper
The independence of the Continuum Hypothesis, Proc. Natl. Acad. Sci. USA. 1963 Dec; 50(6): 1143–1148 (PMC221287)
begins by invoking the Lowenheim-Skolem theorem to assert the ...

**4**

votes

**0**answers

155 views

### Equivalence of Rathjen's continuum hypothesis and another form of the CH without choice

(This question is already posted on Math SE but it isn't answered, so I ask same question on this site.)
The following form of a continuum hypothesis occurs in Rathjen's paper "Indefiniteness in semi-...

**41**

votes

**1**answer

2k views

### Hilbert's alleged proof of the Continuum Hypothesis in “On the Infinite”

As is known, Hilbert attempted a proof sketch of the Continuum Hypothesis in the latter part of his paper, "On the Infinite". It is also known that it is false.
Has there ever been a published ...

**9**

votes

**3**answers

1k views

### Does anyone understand this comment about the continuum hypothesis?

At 31:37 in his lecture titled What is a manifold? posted on Youtube, Mikhail Gromov states that if we do not allow generic functions to exist then the continuum hypothesis is "obviously" true, and ...

**2**

votes

**1**answer

437 views

### Well-ordering of power set of $\omega$

Assuming initially the background set theory ZF, what is the exact status of the existence of a well-ordering on the power set of $\omega$? How much needs to be added to guarantee this?

**3**

votes

**1**answer

336 views

### A Question on HOD, V and GCH

The theorem 1.1 of the following paper:
Mohammad Golshani, V, HOD, and the GCH, Journal of Symbolic Logic.
states that:
Theorem: Assume $V\models ZFC+GCH+~\text{There exists a}~(\kappa+4)-\...

**6**

votes

**1**answer

353 views

### Forcing the negation of CH without adding Cohen reals over L

Suppose CH + "there are no Cohen reals over L". Can we force the negation of CH without adding any Cohen real over L?

**2**

votes

**1**answer

175 views

### Cardinalities of maximal towers in ${\cal P}(\omega)$

For $A,B\subseteq \omega$ we write $A \subseteq^* B$ if $A\setminus B$ is finite. We call ${\cal T}\subseteq {\cal P}(\omega)$ a tower if it is linearly quasiordered with respect to $\subseteq^*$.
...

**17**

votes

**1**answer

586 views

### Can we find CH in the analytical hierarchy?

Recently I was talking to my friend and I have mentioned to him that it was proven that CH is not provably (over ZFC) equivalent to any statement in second-order arithmetic. However, today I found out ...

**1**

vote

**2**answers

382 views

### A question regarding the relation between Freiling's Axiom of Symmetry and real-valued measurable cardnals

A major argument against Freiling's Axiom of Symmetry is the following (this from the wikipedia article of the same name):
"The naive probabalistic notion used by Freiling tacitly assumes that there ...

**13**

votes

**3**answers

900 views

### Complete resolutions of GCH

Let's say that a "complete resolution of GCH" is a definable class function $F: \operatorname{Ord}\longrightarrow \operatorname{ Ord}$ such that $2^{\aleph_\alpha} = \aleph_{F(\alpha)}$ for all ...

**10**

votes

**3**answers

1k views

### Difference between ZFC and ZF+GCH

I hear that the axiom of choice (AC) derives from
The generalized continuum hypothesis(GCH).
And also hear that both AC and GCH are independent of
Zermelo–Fraenkel set theory(ZF).
So, I'm just ...

**11**

votes

**3**answers

616 views

### The continuum hypothesis for packing shapes without overlapping

Consider the finite cross $C$ (=union of line segments $\overline{(0, -1)(0, 1)}$ and $\overline{(-1, 0)(1, 0)}$) and the unit half-circle $H$. It is easy to see that we may pack continuum-many ...

**15**

votes

**1**answer

1k views

### How much of GCH do we need to guarantee well-ordering of continuum?

It's well known that, if GCH holds, then every cardinal can be well-ordered. However, I'm sure we don't need full power of GCH to prove it for specific cardinal, e.g. continuum. I have been wondering, ...

**14**

votes

**0**answers

299 views

### Are there additive subgroups of reals of dimension 1 with no subgroups of dimension strictly between 0 and 1?

I will use $dimA$ to denote the Hausdorff dimension of a set $A \subseteq \mathbb{R}$. Being a null set means having Lebesgue measure zero.
In the 1966 paper "Additive gruppen mit vorgegebener ...

**11**

votes

**2**answers

838 views

### Last Status of Feferman's Conjecture on Indefinite Value of Continuum

The "true" value of $2^{\aleph_0}$ is one of the most fundamental open questions of mathematics and its philosophy. Hundreds of set theoretic results during the last century don't say anything more ...

**12**

votes

**1**answer

482 views

### Does small forcing preserve CH?

Suppose CH holds and $\mathbb{P}$ is a poset of size $\omega_1$, such that forcing with $\mathbb{P}$ preserves $\omega_1$. Does forcing with $\mathbb{P}$ preserve CH? If $\mathbb{P}$ is proper then ...

**10**

votes

**1**answer

438 views

### Ground Axiom and behaviors of continuum function

The Ground Axiom ($GA$) is the assertion that the universe of
sets ($V$) is not a forcing extension of any inner model $W$ by nontrivial forcing
$P\in W$.
Is $GA$ consistent with any possible ...

**7**

votes

**3**answers

1k views

### PFA: A New Godel's Program & A New Large Cardinal Ladder (Updated)

We know $PFA$ implies $2^{\aleph_0}=\aleph_2$.
Q1. What does $PFA$ say about other values of continuum function? Does proper forcing axiom carry any further information about values of continuum ...

**9**

votes

**3**answers

893 views

### Very Large Cardinal Axioms and Continuum Hypothesis

Are very large cardinal axioms like $I_0$, $I_1$, $I_2$ consistent with $CH$ and $GCH$?

**12**

votes

**1**answer

655 views

### Families of pairwise incomparable subsets of the integers

Certain maximal objects whose existence follows from Zorn's Lemma have received some
set-theoretic attention.
Examples are maximal independent families and maximal almost disjoint families.
There is ...

**5**

votes

**1**answer

195 views

### Intermediate submodels and the continuum hypothesis

Let $V$ be a model of $ZFC+GCH$ and let $V[G]$ be a generic extension of $V$ in which $CH$ fails.
Question 1. Is there a model $W$ such that:
1) $V \subseteq W \subseteq V[G],$
2) $W\models CH,$
...

**4**

votes

**3**answers

566 views

### Minimal Generalized Continuum Hypothesis & Axiom of Choice

It is well known that working in the frame of $\text{ZF}$, the Generalized Continuum Hypothesis ($\text{GCH}$) implies the Axiom of Choice ($\text{AC}$), i.e. $\text{ZF}+\text{GCH}\vdash \text{AC}$.
...

**16**

votes

**4**answers

2k views

### A New Continuum Hypothesis (Revised Version)

Define $N_n$ as $n$ th natural number: $N_0=0, N_1=1, N_2=2, ...$.
What happens after exponentiation?
We have the following equation: $2^{N_n}=N_{2^{n}}$.
(Which says: For all finite cardinal $n$ ...

**5**

votes

**1**answer

275 views

### An explicit construction of reals added after some forcing notions

Consider the forcing notion(s) introduced by Friedman (or Mitchell or Neeman) for adding a club subset of $\omega_2$ by finite conditions. In the generic extension CH fails, but I can't see the reals ...

**6**

votes

**2**answers

547 views

### The First Failure of GCH in Large Cardinals Smaller than Measurables

A well known theorem by Scott says:
If $\kappa$ is a measurable cardinal and $\mu$ a normal measure on it and $\mu (\lbrace\lambda\in\kappa~|~2^{\lambda}=\lambda^{+}\rbrace)=1$ then $2^{\kappa}=\...

**8**

votes

**0**answers

322 views

### Ultrapowers of Banach spaces without the continuum hypothesis

Let $\mathcal{U}$ be a non-trivial ultrafilter on the set of integers $\mathbb{N}$, and let $C(K)$ denote the Banach space of continuous functions on a compact $K$. Under the continuum hypothesis CH, ...

**15**

votes

**2**answers

744 views

### Open coloring axiom vs. CH

Is there a simple, direct proof that the open coloring axiom contradicts CH (straight from the definitions, no machinery allowed)? The separable metric spaces version of OCA, if that helps.
Edit: ...

**16**

votes

**2**answers

652 views

### Intersection of compact sets in the unit interval

Let $\mathscr K$ be an uncountable set such that every $K\in\mathscr K$ is a compact subset of $[0,1]$ with positive Lebesgue measure. Does it then follow that there exists an uncountable $\mathscr A\...

**6**

votes

**1**answer

306 views

### The Ground Axiom for special statements of set theory

The Ground Axiom (GA), introduced by Hamkins and Reitz, asserts that the universe is not a nontrivial set forcing extension of any inner model, and it is known that GA is consistent relative to ZFC. ...

**8**

votes

**1**answer

417 views

### Making all cardinals countable and its HOD

Suppose that $G$ is $Col(\omega, <Ord)$-generic over $V$ and let $W=HOD^{V[G]}$. Is $CH$ true in $W$? In general, what can we say about the behaviour of the power function in $W$?
Update. Are the ...

**16**

votes

**2**answers

684 views

### Why does CH imply that there is a unique ultrapower of $\mathbb{N}$?

I've read these words: "How many ultra products $∏_Uℕ$ exist up to isomorphism, where $U$ is a non-principal ultrafilter over $ℕ$? If continuum hypothesis(CH) holds, then obviously just one ..."
i ...

**12**

votes

**3**answers

1k views

### When was the continuum hypothesis born?

The question Solutions to the Continuum Hypothesis states that the continuum hypothesis was posed by Cantor in 1890. In http://en.wikipedia.org/wiki/Continuum_hypothesis the year 1878 is quoted ...

**10**

votes

**1**answer

439 views

### Conceptual Structuralism and Continuum hypothesis

In Ferefman's paper 'Is the Continuum Hypothesis a definite mathematical problem?', he argues that within the philosophy of conceptual structuralism, the continuum hypothesis is not a definite ...

**5**

votes

**2**answers

1k views

### Are all models of ZF + DC + “All set of reals are lebesgue measurable” also models of CH? [duplicate]

Possible Duplicate:
Lebesgue Measurability and Weak CH
I have studied a little set theory and I found that Solovay constructed a model of ZF+DC+"All set of reals are Lebesgue measurable" and I ...