All Questions
Tagged with forcing continuum-hypothesis
24 questions
20
votes
4
answers
3k
views
A New Continuum Hypothesis (Revised Version)
Define $N_n$ as $n$ th natural number: $N_0=0, N_1=1, N_2=2, ...$.
What happens after exponentiation?
We have the following equation: $2^{N_n}=N_{2^{n}}$.
(Which says: For all finite cardinal $n$ ...
user42090
12
votes
1
answer
601
views
Does small forcing preserve CH?
Suppose CH holds and $\mathbb{P}$ is a poset of size $\omega_1$, such that forcing with $\mathbb{P}$ preserves $\omega_1$. Does forcing with $\mathbb{P}$ preserve CH? If $\mathbb{P}$ is proper then ...
- 18.6k
12
votes
1
answer
504
views
Ground Axiom and behaviors of continuum function
The Ground Axiom ($GA$) is the assertion that the universe of
sets ($V$) is not a forcing extension of any inner model $W$ by nontrivial forcing
$P\in W$.
Is $GA$ consistent with any possible ...
user47697
11
votes
1
answer
408
views
The Parovichenko cardinal, is it equal to $\max\{\aleph_2,\mathfrak p\}$?
Let us define the Parovichenko cardinal $\mathfrak{P}$ as the largest cardinal $\kappa$ such that each compact Hausdorff space $K$ of weight $w(K)<\kappa$ is the continuous image of the remainder $...
- 41.9k
11
votes
0
answers
490
views
$\Sigma^2_1$ and the Continuum Hypothesis
This is a follow up to Will Brian's answer to this recent question. In particular, quoting Brian:
"In fact, Paul Larson has pointed out to me that the statement "$\phi$ and $\phi^{-1}$ are conjugate"...
- 7,125
9
votes
0
answers
388
views
On the role of $\diamondsuit$
The well-known axiom $\diamondsuit$ states that there is a sequence $\langle A_\alpha:\alpha<\omega_1\rangle$ (a $\diamondsuit$-sequence) of countable sets with the property that for any $A\...
- 7,125
8
votes
1
answer
507
views
Making all cardinals countable and its HOD
Suppose that $G$ is $Col(\omega, <Ord)$-generic over $V$ and let $W=HOD^{V[G]}$. Is $CH$ true in $W$? In general, what can we say about the behaviour of the power function in $W$?
Update. Are the ...
- 32.1k
7
votes
3
answers
2k
views
PFA: A New Godel's Program & A New Large Cardinal Ladder (Updated)
We know $PFA$ implies $2^{\aleph_0}=\aleph_2$.
Q1. What does $PFA$ say about other values of continuum function? Does proper forcing axiom carry any further information about values of continuum ...
user45878
7
votes
2
answers
896
views
Iterated forcing and CH
I need some help with this theorem: if $P_\beta=\langle P_\alpha,\dot{Q}_\alpha:\alpha\leq\beta\rangle$, $\beta<\omega_2$, is a CSI of proper forcings, $P_\alpha\Vdash \lvert \dot{Q}_\alpha\rvert\...
- 71
6
votes
2
answers
797
views
The First Failure of GCH in Large Cardinals Smaller than Measurables
A well known theorem by Scott says:
If $\kappa$ is a measurable cardinal and $\mu$ a normal measure on it and $\mu (\lbrace\lambda\in\kappa~|~2^{\lambda}=\lambda^{+}\rbrace)=1$ then $2^{\kappa}=\...
user42090
6
votes
1
answer
626
views
Can $H_{\omega_1}$ and $H_{\omega_2}$ be in bi-interpretation synonymy?
This question concerns the possibility of the bi-interpretation synonymy of the structure
$\langle
H_{\omega_1},\in\rangle$, consisting of the hereditarily countable sets, and the structure $\langle ...
- 236k
6
votes
1
answer
490
views
Forcing the negation of CH without adding Cohen reals over L
Suppose CH + "there are no Cohen reals over L". Can we force the negation of CH without adding any Cohen real over L?
- 133
6
votes
1
answer
428
views
The Ground Axiom for special statements of set theory
The Ground Axiom (GA), introduced by Hamkins and Reitz, asserts that the universe is not a nontrivial set forcing extension of any inner model, and it is known that GA is consistent relative to ZFC. ...
- 32.1k
5
votes
1
answer
203
views
Uniformization of almost disjoint families
Suppose $\mathcal{F} \subseteq \mathcal{P} (\omega) $ is an almost disjoint family and $\aleph_0 < \vert \mathcal{F} \vert = \kappa < 2^{\aleph_0} $. Is it consistent that for some such cardinal ...
- 505
5
votes
1
answer
239
views
Intermediate submodels and the continuum hypothesis
Let $V$ be a model of $ZFC+GCH$ and let $V[G]$ be a generic extension of $V$ in which $CH$ fails.
Question 1. Is there a model $W$ such that:
1) $V \subseteq W \subseteq V[G],$
2) $W\models CH,$
...
- 32.1k
5
votes
1
answer
293
views
Does strict order-preservation of powerset curtail the candidates for violation of CH?
Thus, let $\mathrm{OPP}$ be the axiom that $|A|\lt|B| \Rightarrow |2^A|\lt|2^B|$ for any sets $A$ and $B$; and, for any ordinal $\alpha$, let $\mathrm{CH}_\alpha$ be the hypothesis that $\aleph_\alpha=...
- 2,437
4
votes
2
answers
753
views
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
4
votes
1
answer
533
views
How to settle the Generalized Continuum Hypothesis when there are urelements?
Work in $\sf ZFCA$ and permutation models has preceded forcing by several decades. Was it used to settle the question of the Generalized Continuum Hypothesis $\sf GCH$ when urelements are admitted? I ...
- 11.3k
4
votes
1
answer
354
views
An explicit construction of reals added after some forcing notions
Consider the forcing notion(s) introduced by Friedman (or Mitchell or Neeman) for adding a club subset of $\omega_2$ by finite conditions. In the generic extension CH fails, but I can't see the reals ...
- 32.1k
4
votes
0
answers
208
views
PFA for cardinal preserving forcing notions and the CH
Let $FA_{\aleph_1}$(cardinal preserving proper forcings) be the forcing axiom: if $\mathbb{P}$ is a cardinal preserving proper forcing notion and $(D_\xi)_{\xi<\omega_1}$ are dense subsets of $\...
- 32.1k
3
votes
1
answer
447
views
A Question on HOD, V and GCH
The theorem 1.1 of the following paper:
Mohammad Golshani, V, HOD, and the GCH, Journal of Symbolic Logic.
states that:
Theorem: Assume $V\models ZFC+GCH+~\text{There exists a}~(\kappa+4)-\text{...
3
votes
0
answers
113
views
Hereditarily Lindelöf spaces with density continuum
Since there are L-spaces (provably in ZFC), under CH we have regular, hereditarily Lindelöf spaces with density continuum. However, I cannot find an example of such a space under not CH, nor a proof ...
- 31
2
votes
1
answer
390
views
Strong Total Failures vs. Weak Instances of the Generalized Continuum Hypothesis
The exponentiation operator inflicts a subtle information loss on the transfinite numerical equations, pretty similar to the case of $a^2=b^2 \nRightarrow a=b$ in real numbers. In fact, for the ...
2
votes
1
answer
235
views
Continuum function maximum
Easton's theorem can give a very weak nontrivial constraint on continuum function, but it does not hold for singular cardinals. So:
What are the non-trivial constraints on continuum function in ...
- 695