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Results tagged with continuum-hypothesis
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user 11145
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.
7
votes
Accepted
Continuum hypothesis and cardinality of infinite tree paths
The set of all branches is a closed set of reals. Cantor proved that closed sets are either countable or of size continuum.
- 18.6k
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_\alpha …
- 18.6k
4
votes
Strong Total Failures vs. Weak Instances of the Generalized Continuum Hypothesis
In the Foreman-Woodin model (mathscinet MR1087344), for all $\kappa$, $\beth_n(\kappa)$ is at least the $n^{th}$ weakly inaccessible above $\kappa$. Furthermore, this model satisfies the following:
…
- 18.6k
6
votes
PFA: A New Godel's Program & A New Large Cardinal Ladder (Updated)
If we start with an indestructibly supercompact cardinal $\kappa$, then we can do Easton forcing to get any reasonable continuum function on the regular cardinals $\geq \kappa$ and retain the supercom …
- 18.6k
11
votes
Accepted
A question regarding the relation between Freiling's Axiom of Symmetry and real-valued measu...
Real-valued measurable cardinals (RVM) are equiconsistent with 2-valued measurable cardinals. I believe this is due to Solovay and Kunen. (Solovay for the forcing direction, Kunen for the inner mode …
- 18.6k