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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.
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
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 …
- 6,367
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, witho …
- 236k
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
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 …
- 236k
23
votes
Accepted
Is it consistent with ZFC that the real line is approachable by sets with no accumulation po...
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 y …
- 7,817
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 …
- 236k
8
votes
Accepted
How to settle the Generalized Continuum Hypothesis when there are urelements?
In ZFCA, every set is equinumerous with a pure set, since every well ordering of a set makes it bijective with an ordinal, which is pure. Therefore the cardinal structure of any model of ZFCA is refle …
- 236k
31
votes
Why does CH imply that there is a unique ultrapower of $\mathbb{N}$?
The point is that the ultrapower of any structure $\mathcal{M}$ by
a nonprincipal ultrafilter $\mu$ on $\mathbb{N}$ is countably
saturated, that is, it realizes any finitely satisfiable $n$-type with …
- 236k
7
votes
Is "the purely probabilistic version of Freiling's axiom of symmetry" disprovable in ZFC?
This is not a full answer, but let me just point out that the statement is relatively consistent both with CH and also with $\neg$CH.
CH implies the statement, since we can take $S$ to be the (convers …
- 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 …
- 236k
174
votes
Accepted
Solutions to the Continuum Hypothesis
Since you have already linked to some of the contemporary
primary sources, where of course the full accounts of those
views can be found, let me interpret your question as a
request for summary accoun …
- 4,713
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
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