All Questions
Tagged with forcing inner-models
8 questions
22
votes
2
answers
1k
views
Gently changing measure
This question was asked and bountied on MSE without answer, so I'm porting it here:
There's an easy way to change the measure of a set of reals by moving to a larger universe: simply make $\mathbb{R}$...
- 20.5k
6
votes
1
answer
242
views
Generic saturation of inner models
Say that an inner model $M$ of $V$ is generically saturated if for every forcing notion $\Bbb P\in M$, either there is an $M$-generic for $\Bbb P$ in $V$, or forcing with $\Bbb P$ over $V$ collapses ...
- 39.8k
9
votes
1
answer
228
views
Is there a minimal extension of $L$ that is not a forcing extension?
It's well known that Sacks forcing constructs a real of minimal constructability degree, i.e. a real $x$ such that for any $y\in L(x) \setminus L$, $L(y) = L(x)$. It's also well known that certain ...
- 12.5k
6
votes
1
answer
571
views
fake and weak cardinals
Suppose $\lambda$ is a successor of a singular cardinal. We will say $\lambda$ fake if there is a transitive set $M$ such that $\lambda \subseteq M$ satisfying $\mathrm{ZFC}^-$ (ZFC without powerset) ...
- 18.6k
10
votes
4
answers
554
views
What are some kinds of models where DC holds?
There are a lot of ways to build a model where DC fails. However, all of them that I'm aware of involve adding at least a messy set of reals (or rather, taking a forcing extension and then passing to ...
- 20.5k
7
votes
1
answer
293
views
Extending Sacks forcing
Sacks forcing allows us to build a model $V[G]$, such that there is no "intermediate model" between $V$ and $V[G]$, meaning if $V \subseteq W \subseteq V[G]$ is a model of ZFC then either $W = V$ or $...
- 529
10
votes
1
answer
590
views
Singular in $V$ regular in $HOD$
Prikry forcing can be used to produce a model $V$ of $ZFC$ such that fo rsome cardinal $\kappa$ we have:
(1) $\kappa$ is singular in $V$ of cofinality $\omega,$
(2) $\kappa$ is regular (and in fact ...
- 32.1k
7
votes
0
answers
262
views
$V$ as a $HOD$ of its class generic extension
By an old result of Roguski, The theory of the class $HOD$, any model $V$ of $ZFC$ has a class generic extension $V[G]$ such that $HOD$ of $V[G]$ equals $V$.
This result is also stated and generalized ...
- 32.1k