47
votes
Accepted
Hilbert's alleged proof of the Continuum Hypothesis in "On the Infinite"
The article "Hilbert and Set Theory" by Dreben and Kanamori devotes Section 7 to this argument and an analysis of its flaws. Dreben and Kanamori use the translation provided by van Heijenoort, so that ...
- 21.2k
45
votes
Accepted
What is the most "concrete-feeling" equivalent formulation of the Continuum Hypothesis that you can think of?
Since we can force $CH$ and $\neg CH$ between transitive models of ZFC, we know that $CH$ cannot be equivalent to a statement that is absolute between transitive models, including arithmetic ...
43
votes
What is the most "concrete-feeling" equivalent formulation of the Continuum Hypothesis that you can think of?
Here are a few of my favorite characterizations of the continuum hypothesis:
Sierpiński (1951) proved that CH is equivalent to the assertion that there is a partition of the plane into two sets $\...
37
votes
Accepted
If someone can prove Goldbach conjecture assuming the continuum hypothesis, do we consider the conjecture proved?
Because the Goldbach conjecture is an arithmetic statement, it is absolute between any two models which agree on the natural numbers.
Now, given any model of $\sf ZFC$, $M$, there is a forcing ...
- 39.7k
29
votes
Accepted
On the probability of the truth of the continuum hypothesis
For any $E$ modelling ZFC and for each $n\in\omega$, we must have $(n,n)\not\in E$. Therefore $M_{ZFC}$ is contained in the cylinder set defined by $(0,0)\not\in E,\dots,(n,n)\not\in E$ which has ...
- 28.2k
27
votes
What is the most "concrete-feeling" equivalent formulation of the Continuum Hypothesis that you can think of?
Theorem (Erdős [1964]). The following assertion is equivalent to $\mathsf{CH}$: There exists an uncountable family $ℱ$ of entire analytic functions such that for each $z ∈ ℂ$, the set of values $\{ f(...
27
votes
What is the most "concrete-feeling" equivalent formulation of the Continuum Hypothesis that you can think of?
A cloud around $x$ in $\Bbb R^2$ is a set $C$ such that for all lines $\ell$ through $x$, $C\cap\ell$ is finite. A cloud is a cloud around some point.
Theorem: (Komjáth, Schmerl) the continuum ...
26
votes
Unnecessary uses of the Continuum Hypothesis
Theorem: The space $\mathbb N^*$ of non-principal ultrafilters on $\mathbb N$ is not homogeneous.
Using CH, it is fairly straightforward to prove there is a special kind of ultrafilter called a $P$-...
26
votes
What is the most "concrete-feeling" equivalent formulation of the Continuum Hypothesis that you can think of?
Morayne showed the following in this paper:
Theorem. The continuum hypothesis is equivalent to the existence of a surjection $F : \mathbb{R} \to \mathbb{R}^2$ such that for every $x \in \mathbb{R}$, ...
25
votes
Accepted
Does $V = \textit{Ultimate }L$ imply GCH?
In his slide Absolutely ordinal definable sets John Steel writes:
At the same time, one hopes that V = ultimate L will yield a detailed fine structure theory for V, removing the incompleteness that ...
- 32.1k
24
votes
Accepted
Must uncountable standard models of ZFC satisfy CH?
Remarks (2) and (3) are added in this edit.
What Cohen's quoted proof outline is leaving implicit is the following statement in which $\mathrm{Con}(T)$ means "$T$ is consistent".
$(*)$ ...
- 17.7k
24
votes
CH in non-set theoretic foundations
In ETCS, the continuum hypothesis can be encoded as the statement that given a set $X$ with injections $\mathbb{N} \hookrightarrow X \hookrightarrow 2^{\mathbb{N}}$, then there is either an ...
- 35.4k
23
votes
Accepted
Can GCH fail everywhere every way?
No. An early nontrivial constraint on the $\beth$ function comes from Kőnig's Theorem, that for all infinite $\kappa$, $\mathrm{cf}(2^\kappa)>\kappa$. This implies that we cannot have $\beth_\...
- 18.6k
23
votes
Accepted
Is it consistent with ZFC that the real line is approachable by sets with no accumulation points?
It is a very nice question, but unfortunately, this is impossible.
Each member $s\in S$ must be countable, since uncountable sets have accumulation points. And since the hierarchy is accumulating as ...
- 236k
20
votes
Accepted
Is $[0,1]$ a disjoint union of $\aleph_1$ compact subsets with empty interior?
It is independent of ZFC. As you mention, CH implies the answer is yes. A different axiom, MA+$\neg$CH, implies the answer is no.
A little more precisely, there is a cardinal number denoted $\mathrm{...
- 18.5k
18
votes
Is it possible to define cardinals that are distinct from either the $\aleph$ numbers or $\beth$ numbers?
One can consistently generate new cardinals simply by combining the two methods you have mentioned.
For example, I claim that it is relatively consistent with ZF that the cardinal $\aleph_1+\beth_1$ ...
- 236k
17
votes
CH in non-set theoretic foundations
One consideration will be that the CH statement can become ambiguous in weaker foundations, since statements that are equivalent in ZFC are not always equivalent in weaker theories.
For example, ...
- 236k
16
votes
How many well-orders of reals are there?
Every well order of the real numbers has some order type between $\mathfrak{c}$ and $\mathfrak{c}^+$, and for any given order type arising, every permutation of $\mathbb{R}$ induces another well order ...
- 236k
15
votes
Accepted
Almost everywhere “filling” of the continuum by the first uncountable cardinal without CH
The existence of such an $f$ is consistent with $\mathsf{ZFC+\neg CH}$. Suppose $\mathfrak{c}=\aleph_2$ and every set of size $\le\aleph_1$ is null (this is consistent with $\mathsf{ZFC}$; it follows, ...
- 21.2k
14
votes
Accepted
Continuum Hypothesis and the fact that every co-finite topological space, with uncountable underlying set , is contractible
Nice question!
I claim that this property does not necessarily imply CH. As Todd
guessed in his comment, the answer is related to certain cardinal
characteristics of the continuum.
Specifically, let ...
- 236k
14
votes
Accepted
Can the cardinal $2^{\aleph_0}$ be order-embedded in ${\cal P}(\omega)/(\text{fin})$?
Yes! The fact that this is consistent is originally due to Laver. Later, Baumgartner, Frankiewicz, and Zbierski strengthened Laver's result to the following theorem:
Theorem: Is it consistent that $\...
- 18.5k
14
votes
What is the most "concrete-feeling" equivalent formulation of the Continuum Hypothesis that you can think of?
The continuum hypothesis is equivalent to the assertion that for any set $X\subseteq\mathbb R$ either White or Black has a winning strategy in the following infinitely long game $G(X)$.
At the $n^\...
14
votes
What is the most "concrete-feeling" equivalent formulation of the Continuum Hypothesis that you can think of?
This is a Hilbert-hotel variant I came up with that I'm putting in a philosophy of math book I'm currently writing.
In the infinite binary tree shown, imagine that it is a hotel, and infinitely many ...
13
votes
What is the most "concrete-feeling" equivalent formulation of the Continuum Hypothesis that you can think of?
My favorite equivalence is that CH fails if and only if whenever we color $\mathbb{R}$ with countably many colors, we can find monochromatic (and distinct) $x$, $y$, $u$, and $v$ such that $x+y = u + ...
13
votes
Is it consistent with ZFC that the real line is approachable by sets with no accumulation points?
Here's a ZF proof that if $S$ is a chain of sets with $\bigcup S = \mathbb{R},$ then there is $X \in S$ which contains a countable set dense in some nonempty open set.
If there is $X \in S$ such that $...
- 4,316
12
votes
Can GCH fail everywhere every way?
When working in ZF, one can have more freedom. See
An Easton-like Theorem for Zermelo-Fraenkel Set Theory with the Axiom of Dependent Choice and An Easton-like theorem for Zermelo-Fraenkel Set Theory ...
- 32.1k
12
votes
Uniqueness results that follow from CH
An example would be the uniqueness of the asymptotic cone for uniform lattices in $SL_m(\mathbb R)$. Under $\neg CH$, these admit $2^{2^{\aleph_0}}$ non-homeomorphic asymptotic cones.
See
Asymptotic ...
12
votes
Uniqueness results that follow from CH
Under CH, we have saturated models of size continuum of any consistent first-order theory in a countable language, and for a complete theory these are unique by the back-and-forth method.
(In my paper,...
11
votes
Is it still an open problem whether $\mathbb{R}^\omega$ is normal in the box topology?
I'm transcribing here some things from the comments section, so that this question can be marked as answered:
The problem is still open.
This survey by Roitman and Williams, from 2015, said that it ...
11
votes
Accepted
Bernstein's proof of the continuum hypothesis
This is really a long comment. This paper has been reviewed twice by zbMATH: one by H. B. Curry, which is not informative; another by W. Ackermann, which is in German. The following is the (...
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