All Questions
Tagged with forcing cardinal-arithmetic
11 questions
15
votes
1
answer
615
views
Changing the cofinality of a regular cardinal without collapsing any cardinals?
I have a short but hopefully interesting question on cardinal arithmetic and collapsing cardinals:
Is it possible to change the cofinality of a regular cardinal without collapsing any cardinals?
Is ...
- 307
8
votes
1
answer
260
views
Can we force $\aleph_\omega^\omega<2^{\aleph_\omega}$?
Since $\operatorname{cf}(\aleph_\omega)=\omega$, $\aleph_\omega<\aleph_\omega^\omega$. However, can we force $\aleph_\omega^\omega<2^{\aleph_\omega}$? I am especially interested in models for ...
- 2,206
5
votes
0
answers
205
views
Does this proof by Shelah use any "hidden assumptions"?
Recall that the approachability ideal for $\kappa$, denoted $I[\kappa]$, consists of all sets $A\subseteq\kappa$ such that there is a sequence $\overline{a}=(a_{\alpha})_{\alpha\in\kappa}$ of bounded ...
- 1,799
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
4
votes
0
answers
182
views
Generic two-cardinal behavior of first-order sentences
This is a hopefully improved version of a question I asked before and then deleted because it was based on some fundamentally incorrect assumptions.
Some first-order theories are able to control the ...
- 12.5k
17
votes
0
answers
558
views
Gitik's work on Shelah's weak hypothesis
It seems that Moti Gitik has recently refuted some variants of Shelah's weak hypothesis. For this see the title and abstract of his talk at the Set Theory, Model Theory and Applications conference.
I ...
- 32.1k
7
votes
1
answer
374
views
Does Laver Forcing add an infinitely often equal real?
Given a model $V$ of set theory and an inner model $W \subseteq V$, a real $x \in V \cap \omega^\omega$ is infinitely often equal over $W$ if for each real $y \in \omega^\omega$ there are infinitely $...
- 1,882
9
votes
2
answers
540
views
Reals which must, can't or might be added by forcing
Let $W \subseteq V$ be an inner model of ZFC. There are a variety of theorems that characterize when a real $x \in V$ is the generic of a forcing notion $\mathbb P \in W$, for example, the ...
- 1,882
7
votes
2
answers
248
views
Getting PFA + GCH above $\omega$
The Proper Forcing Axiom kills CH in a particularly specific way: it implies that $2^{\aleph_0}=\aleph_2$. However, its impact on the continuum function above $\aleph_0$ is much less clear. It is ...
- 21.2k
17
votes
1
answer
1k
views
When can power sets be limit cardinals?
My original question (posted in here at the Math.SE) was:
Is it possible to create a model of ZFC, so that the cardinality of each power set is a limit cardinal (as opposed to GCH where they are ...
- 529
5
votes
2
answers
448
views
Prevalent singular cardinals hypothesis
The following notion is introduced by Assaf Rinot:
Definition. A singular cardinal $\kappa$ is a prevalent singular
cardinal iff there exists a family $\mathbb{A}\subset P(\kappa)$ with $|\mathbb{A}...
- 32.1k