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Tagged with axiom-of-choice continuum-hypothesis
10
questions
4
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832
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About the relationship between the generalized continuum hypothesis and the axiom of choice
I was trying to get a short, intuitive proof of Sierpinski’s theorem (gch implies axiom of choice) and I could but only by using the following gch2 for the generalized continuum hypothesis gch.
gch: ...
- 483
1
vote
1
answer
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Is existence of a cardinal that witness non-failure of GCH everywhere everyway, a theorem of ZF?
In an earlier positing to $\mathcal MO$, it appears that the answer to if the $\sf GCH$ can fail everywhere in every way is to the negative, this is the case in $\sf ZFC$, however it also appears that ...
- 10.2k
10
votes
1
answer
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Does GCH for alephs imply the axiom of choice?
GCH for alephs means the statement that, for any aleph $\kappa$, there are no cardinals $\mathfrak{r}$ such that $\kappa<\mathfrak{r}<2^\kappa$.
Does GCH for alephs imply the axiom of choice?
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- 1,563
4
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0
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Equivalence of Rathjen's continuum hypothesis and another form of the CH without choice
(This question is already posted on Math SE but it isn't answered, so I ask same question on this site.)
The following form of a continuum hypothesis occurs in Rathjen's paper "Indefiniteness in semi-...
- 2,910
3
votes
1
answer
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Well-ordering of power set of $\omega$
Assuming initially the background set theory ZF, what is the exact status of the existence of a well-ordering on the power set of $\omega$? How much needs to be added to guarantee this?
- 15.8k
10
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3
answers
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Difference between ZFC and ZF+GCH
I hear that the axiom of choice (AC) derives from
The generalized continuum hypothesis(GCH).
And also hear that both AC and GCH are independent of
Zermelo–Fraenkel set theory(ZF).
So, I'm just ...
16
votes
1
answer
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How much of GCH do we need to guarantee well-ordering of continuum?
It's well known that, if GCH holds, then every cardinal can be well-ordered. However, I'm sure we don't need full power of GCH to prove it for specific cardinal, e.g. continuum. I have been wondering, ...
- 27.7k
4
votes
2
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Minimal Generalized Continuum Hypothesis & Axiom of Choice
It is well known that working in the frame of $\text{ZF}$, the Generalized Continuum Hypothesis ($\text{GCH}$) implies the Axiom of Choice ($\text{AC}$), i.e. $\text{ZF}+\text{GCH}\vdash \text{AC}$.
...
user42090
5
votes
2
answers
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Are all models of ZF + DC + "All set of reals are lebesgue measurable" also models of CH? [duplicate]
Possible Duplicate:
Lebesgue Measurability and Weak CH
I have studied a little set theory and I found that Solovay constructed a model of ZF+DC+"All set of reals are Lebesgue measurable" and I ...
21
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1
answer
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The Continuum Hypothesis and Countable Unions
I recently edited an answer of mine on math.SE which discussed the implication of the two assertions:
$AH(0)$ which is $2^{\aleph_0}=\aleph_1$, and
$CH$ which says that if $A\subseteq 2^{\omega}$ and ...
- 38.5k