Forcing is a method first used to prove the continuum hypothesis is independent of the classical axioms of set theory
13
votes
2answers
463 views
Woodin's unpublished proof of the global failure of GCH
An unpublished result of Woodin says the following:
Theorem. Assuming the existence of large cardinals, it is consistent that $\forall \lambda, 2^{\lambda}=\lambda^{++}.$
In the paper "The ...
7
votes
1answer
151 views
On free limits of iterations
Shelah isolates the notion of "$\aleph_1$-free iteration" in the first two sections of Chapter IX from Proper and Improper Forcing, and he proves there that properness is preserved by this sort of ...
3
votes
1answer
149 views
Large cardinals and mild extensions
It is known that for many large cardinals $\kappa$ (like weakly compact, measurable,...) , $\kappa$ remains large of the same type after forcings of size $<\kappa$. Now the questions are:
Question ...
8
votes
0answers
202 views
Some questions about $0^{\sharp}$ and forcing over $L$
1-Let $P\in L$ be a nontrivial set forcing or even a tame class forcing (tameness in the sense of Sy Friedman; see for example his Handbook paper), and let $G$ be $P-$generic over $L$. Then it is ...
2
votes
2answers
139 views
Preservation of measurable cardinals in mild extensions
I would like some help with the proof of the preservation of measurable cardinals in mild extensions, as I am a bit new with forcing.
By mild extensions, I mean the generic extension produced from a ...
5
votes
2answers
204 views
Failure of diamond at large cardinals
What is known about the failure of $\diamond_{\kappa}$ (diamond at $\kappa$) for $\kappa$ (the least) inaccessible, (the least) Mahlo and (the least) weakly compact.
4
votes
1answer
327 views
set theory forcing
Suppose $M$ satisfies the $CH$ and that we force over $M$ with $\mathbb{P}=Fn(I,2)$ where $(\omega_{2} \leq |I|))^{M}$, that is, with the finite partial functions from $I$ to $2$. If $f \in M[G] \cap ...
6
votes
0answers
161 views
Existence of a regular subposet which collapses everything except the top cardinal
Suppose $\delta$ is an inaccessible cardinal, and $\mathbb{P}$ is the Levy Collapse $\text{Col}(\kappa, \delta)$ which adds a surjection from $\kappa \to \delta$ (for some regular $\kappa < ...
3
votes
1answer
201 views
Trivial forcings which are not very trivial
Suppose that $M$ is a model of $\sf ZFC$, and we add some generic set $G$. Then it is not hard to see that for every $x\in M[G]$ it holds $M\subseteq M[x]\subseteq M[G]$.
Given $x\in M[G]$ such that ...
6
votes
3answers
251 views
Class forcing: Pelletier vs Friedman
[Apologies in advance for a fluffy question]
I'm reading this old paper by Pelletier, where he gives a Boolean-valued model version of class forcing, assuming that the Boolean algebra in question can ...
4
votes
1answer
106 views
Amalgamation of two ccc algebras may collapse the continuum
The claim that appears in the title of this question is mentioned in the paper "On Shelah's amalgamation" by Judah and Roslanowski. I'd really like to see a proof of this fact, but unfortunately I ...
5
votes
1answer
258 views
collapsing successor of singular
Let $\lambda$ be a singular cardinal. Is it consistent that there is a forcing of size $\lambda^+$ that collapses $\lambda^+$ while preserving all cardinals below $\lambda$?
(Note that even without ...
5
votes
1answer
187 views
Question about Shelah's version of “Shooting a club” found in PIF
Suppose $S \subset \omega_{1}$ is stationary co-stationary. Then there is a forcing notion $P_{S}$ which shoots a closed unbounded $C \subset S$ without collapsing cardinals (or
changing ...
10
votes
1answer
335 views
Forcing mildly over a worldly cardinal.
A cardinal $\theta$ is worldly if $V_{\theta}$ is a model of ZFC. We could force to collapse $\theta$ to a successor cardinal, for example, and destroy the worldliness of $\theta$, but is there a ...
4
votes
1answer
127 views
Which $\omega_1$-trees are proper?
Consider a tree $(T, <_T)$ of height $\omega_1$, with countable levels. One can view $T$ as a forcing poset by calling a condition $s\in T$ stronger than $t\in T$ if $t <_T s$.
My question is: ...