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Results tagged with continuum-hypothesis
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user 7206
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.
5
votes
Well-ordering of power set of $\omega$
More or less the only you need to require it explicitly.
It is consistent with any amount of Dependent Choice that the reals cannot be well ordered. Just blow the continuum to be large enough, and pre …
- 39.7k
37
votes
Accepted
If someone can prove Goldbach conjecture assuming the continuum hypothesis, do we consider t...
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 extensi …
- 39.7k
15
votes
Accepted
Difference between ZFC and ZF+GCH
In some sense $\sf GCH$ is a limiting axiom. While it solves a lot of things, it also means that certain things we are interested in become false or trivialized. And that's no fun.
For example, forci …
7
votes
The Continuum Hypothesis and Countable Unions
The answer for the second question is no. Truss proved in [1] that if we repeat Solovay's construction from a limit cardinal $\kappa$, we obtain a model in which the following properties:
Countable …
- 39.7k
8
votes
Accepted
What if $\mathbb{R}$ is in bijection with the cardinals less than $\frak{c}$?
Yes. Start with a model of $\sf CH$, then take the least fixed point with uncountable cofinality. Call that $\kappa$. Now add $\kappa$ Cohen reals.
Since fixed points form a club of ordinals, you can …
- 39.7k
26
votes
Accepted
How much of GCH do we need to guarantee well-ordering of continuum?
Yes, you are right. This is a theorem of Specker. If there are no intermediate cardinals between $A,\mathcal P(A)$ and $\mathcal{P(P(}A))$, then $A$ can be well-ordered.
You can find nice details in:
…
- 39.7k
23
votes
When was the continuum hypothesis born?
Finally, a good use for the newly purchased copy of "Zermelo's Axiom of Choice".
Moore writes that Cantor formulated the following problem in 1878:
Every infinite subset of $\Bbb R$ is either den …
- 39.7k
10
votes
Accepted
Does GCH for alephs imply the axiom of choice?
The answer is positive, yes.
Note that $2^\kappa\leq 2^{\kappa^+}$, and therefore $\kappa^+\leq\kappa^++2^\kappa\leq 2^{\kappa^+}$. So either $2^\kappa=2^{\kappa^+}$ or $2^\kappa=\kappa^+$.
In the fir …
- 39.7k
5
votes
Minimal Generalized Continuum Hypothesis & Axiom of Choice
Let me add on Joel's answer and point out that in fact in $\sf ZF$ the following weakening of $\sf GCH$ holds:
For every $A$, if $A$ is well-orderable, then $\mathcal P(A)$ is well-orderable $\imp …
- 39.7k
4
votes
Are all models of ZF + DC + "All set of reals are lebesgue measurable" also models of CH?
Some remarks:
First of all it is known since 1906 that the axiom of choice implies the existence of non-measurable sets. What Solovay have shown is that it is consistent (relative to the existence of …
- 39.7k
3
votes
Accepted
Is existence of a cardinal that witness non-failure of GCH everywhere everyway, a theorem of...
Here's a somewhat trivial answer.
Note that $V_\alpha$, for an infinite $\alpha$, have a particularly nice set of properties which follow from the fact that $|V_\alpha\times V_\alpha|=|V_\alpha|$.
Now …
- 39.7k