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Results tagged with continuum-hypothesis
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user 11115
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.
8
votes
Complete resolutions of GCH
Let me add more examples:
If we consider the global behavior of the power function, then we have for example:
(A) (Foreman-Woodin): $F$ can be such that $F(\alpha)>\alpha+\omega,$ all $\alpha$ ( …
- 32.1k
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
11
votes
Solutions to the Continuum Hypothesis
Let me add in short details, views of three famous set theorists about the problem:
Shelah’s answer: The question was wrong. The right question should be about other
combinatorial objects. There we …
- 32.1k
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 tha …
- 32.1k
4
votes
A Question on HOD, V and GCH
Thanks for referring to my paper. Here are some notes:
1) In the above paper, we have a fixed gap 3 everywhere, while if someone wants a model in which $GCH$ fails everywhere but its $HOD$ satisfies …
- 32.1k
6
votes
The First Failure of GCH in Large Cardinals Smaller than Measurables
As far as I know the best results in this direction are due to Levinski (see his paper "Filters and large cardinals").
The following results are taken from the above paper. First a few definitions:
…
- 32.1k
7
votes
Accepted
PFA: A New Godel's Program & A New Large Cardinal Ladder (Updated)
First note that PFA implies SCH. On the other hand, PFA is indestructible under $\aleph_2-$directed closed forcings, so we can force any pattern of Easton theorem for regular cardinals $\geq \aleph_2. …
- 32.1k
7
votes
Foundational results dependent on/equivalent to the continuum hypothesis or its negation?
As it is stated in the comments, one reference is Sierpinski's book, Hypothese Du Continu, though it is not in English.
Another reference is Propositions Equivalent to the Continuum Hypothesis.
See al …
- 32.1k
7
votes
Mathematical Evidence Backing $|\mathbb{R}|=\aleph_2$
You may also look at Judah's paper Was Godel right.
Judah intensively discusses why the actual evidences accumulated by 30 years of forcing considerations suggest that the most reasonable size for t …
- 32.1k
11
votes
Accepted
A New Continuum Hypothesis (Revised Version)
In the following answer, by Foreman-woodin model, I mean the model constructed by them in the paper "The generalized continuum hypothesis can fail everywhere.
Ann. of Math. (2) 133 (1991), no. 1, 1–35 …
- 32.1k