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Results tagged with continuum-hypothesis
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user 1946
Questions about the continuum hypothesis, or where the continuum hypothesis or its negation plays a role. This tag is also suitable, by extension, to refer to the generalized continuum hypothesis and related issues.
18
votes
Is it possible to define cardinals that are distinct from either the $\aleph$ numbers or $\b...
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
6
votes
Accepted
Cardinalities of maximal towers in ${\cal P}(\omega)$
The answer is yes, for I claim that every maximal chain has size continuum.
Suppose that $C$ is a chain of subsets of $\mathbb{N}$ which is
maximal with respect to almost inclusion. Let's work in the …
- 236k
43
votes
What is the most "concrete-feeling" equivalent formulation of the Continuum Hypothesis that ...
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 $\mat …
8
votes
Freiling's axiom of symmetry and CH - need some help
The continuum hypothesis implies the failure of Freiling's axiom of symmetry and indeed it is equivalent to the failure of this axiom.
To see this, assume first that CH holds. What this means is that …
- 236k
8
votes
Accepted
Is each of the infinite statements of the Generalized Continuum Hypothesis independent?
In the interest of bringing the question to a conclusion, let me say that it is an immediate consequence of Easton's theorem, as mentioned in the comments, that the various GCH assertions at the $\ale …
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
10
votes
Complete resolutions of GCH
$\newcommand\Ord{\text{Ord}}$Easton's theorem allows us to control the continuum function on the infinite regular cardinals, and in particular, on the infinite successor cardinals, in a very flexible …
- 236k
8
votes
Making all cardinals countable and its HOD
$\newcommand\gHOD{\text{gHOD}}
\newcommand\HOD{\text{HOD}}
\newcommand\ZFC{\text{ZFC}}
\newcommand\GCH{\text{GCH}}$
I believe that it was Peter Koepke who first proposed that we
should investigate t …
- 236k
12
votes
Accepted
Ground Axiom and behaviors of continuum function
In the paper The ground axiom is consistent with $V\ne{\rm HOD}$ (J. D. Hamkins, J. Reitz, W.H. Woodin, PAMS 136(8):2008), we prove that the ground axiom is consistent with $V\neq\text{HOD}$, and rema …
- 236k
5
votes
Accepted
Does strict order-preservation of powerset curtail the candidates for violation of CH?
First, let me remark that the particular way that you've posed the question has several problematic issues of formalization. One issue, noted by François, Andres and Andreas, is that it doesn't make s …
- 236k
7
votes
The Ground Axiom for special statements of set theory
In your definition, probably you intend that the forcing is nontrivial, since otherwise any consistent $\Phi$ would qualify via trivial forcing. Note also that your forceable terminology conflicts wit …
- 236k
21
votes
Accepted
When $2^\alpha = 2^\beta$ implies $\alpha=\beta$ ($\alpha,\beta$ cardinals)
François gives the correct affirmative answer. For the negative side, the usual method of proving that the negation of the Continuum Hypothesis is consistent with ZFC is to use the method of forcing t …
- 236k
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 …
5
votes
The continuum hypothesis for packing shapes without overlapping
Since you ask specifically about compact $S$, it is natural also to consider only very nice packings. So let us consider only Borel packings, and in this simplified case, the answer is yes.
Specific …
- 236k
14
votes
Accepted
Continuum Hypothesis and the fact that every co-finite topological space, with uncountable u...
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