All Questions
43
questions
13
votes
2
answers
826
views
Infinite wacky race
Dick Dastardly is taking part in an infinite wacky race. What is infinite about it, you ask? Well, just everything! There are infinitely many racers, every one of which can run infinitely fast and the ...
- 3,881
2
votes
0
answers
51
views
Finite/Countable support iteration of countable cofinality and mad families
Using a finite support iteration $\langle P_\alpha, \dot{Q}_\alpha : \alpha < \kappa\rangle$ it has been shown that if $\kappa$ has uncountable cofinality, then using Mathias forcing one obtains a ...
- 415
7
votes
1
answer
108
views
Is the dominating number of this continuum graph a small cardinal?
Define the relation $\sim$ over $\mathcal C := \{0, 1\}^{\mathbb N}$ given by $(x_n)_n\sim (y_n)_n$ iff $\exists^\infty k: a_{k+i} = b_{k+i},~i=0, 1, \dots, k-1$.
What is the least size of a subset $\...
- 3,881
2
votes
0
answers
64
views
Why doesn't $\prod_n\operatorname{Fn}_{\aleph_n}(\aleph_n,2)$ collapse $\aleph_{\omega+1}$?
Consider the product forcing $\prod_n\text{Fn}_{\aleph_n}(\aleph_n,2)$ with full support, where $\text{Fn}_{\aleph_n}(\aleph_n,2)$ is the poset of partial functions from $\aleph_n$ to $2$ of size less ...
- 555
2
votes
2
answers
113
views
Countably closed forcing that collapses $\mathfrak{c} $ to$\aleph_1$
A paper I'm reading claims that we can find a countably closed forcing notion that collapses $\mathfrak{c}$ to $\aleph_1$, but I can't think of one. I know of the Lévy Collapse, but I don't think that ...
- 566
1
vote
1
answer
95
views
Reducing to regular cardinals in c.c.c. implies same cardinals and cofinalities
In Chapter 14 of Jech's Set Theory, he asserts the following:
Definition 14.33. A forcing notion $P$ satisfies the countable chain condition (c.c.c.) if every antichain in $P$ is at most countable.
...
- 1,453
13
votes
1
answer
501
views
Which ordinals can be "mistaken for" $\aleph_1$?
I've just finished working my way through Weaver's proof of the consistency of the negation of the Continuum Hypothesis in his book Forcing for Mathematicians. One of the key points in this proof is ...
3
votes
1
answer
173
views
Several forcing axioms imply $2^{\aleph_0 }= \aleph_2$. What about $2^{\aleph_1}$?
On the one hand, it seems intuitive that $2^{\aleph_1 }> 2^{\aleph_0}$, because $\aleph_1 > \aleph_0$. However, I also know that, like many things involving the continnum function, that's ...
- 22.9k
1
vote
1
answer
87
views
Is it the property "be unbounded" upwards absulote?
I'm reading the paper Cardinal invariants above the continuum by J. Cummings and S. Shelah. There they have de following lemma:
I have a doubt about with the yellow part: what they really means with ...
- 2,662
6
votes
0
answers
62
views
In Cohen forcing $\mathrm{Fn}(\kappa\times\omega, 2)$ prove $(\lambda^\vartheta)^{M[G]} = ((\max\{\lambda, \kappa\})^\vartheta)^M$ [duplicate]
I'm trying to find a solution for exercise G1 of Chapter VII in Kunen's "Set Theory - An Introduction to Independence Proofs".
The question is in the context of Cohen forcing. We start with ...
- 61
0
votes
1
answer
78
views
Proper and improper forcing Stationarity
In his book on Proper and Improper forcing Shelah writes on page 89:
In Sect. 1 we introduce the property "proper" of forcing notions: preserving
stationarity not only of subsets of $\...
- 3,978
2
votes
0
answers
50
views
a name for a class of cardinals
I'm not a native English speaker; what S in SCar below might stand for ?
- 3,978
1
vote
0
answers
38
views
Not a correct subscript?
In his book Proper and Improper forcing on page $42$ in the proof of Fact 7.3 on line $4$
Shelah says "where ${\cal S_{\leq \aleph_1}}(A)$" but he never uses this expression.
He only uses it ...
- 3,978
2
votes
1
answer
253
views
Forcing $2^{\omega} = \omega_{\omega_1}$ together with $2^{\omega_1} = \omega_{\omega_2}$
I'm studying from Kunen's Set Theory, and I came across this exercise (G6 ch. 7):
Suppose $M$ satisfies $\text{GCH}$. Let $\kappa_1 < \dots < \kappa_n$ be regular cardinals of $M$ and let $\...
- 2,611
2
votes
1
answer
277
views
A lemma in the proof of $\kappa$-chain condition preservation with iterated forcing
I do have a question about a fragment of a proof of a theorem in iterated forcing.
It is the one that Jech invokes in the following form as theorem 16.30 in the 'Seth Theory' book and as Part II, Thm ...
- 167