4
votes
1answer
172 views

About subposet of Levy collapse

Let $\lambda>\kappa$ and $\operatorname{Coll}(\kappa, \lambda)$ be the poset collapsing $\lambda$ to $\kappa$. Pick a subposet $P$ which is $\lt\kappa$-closed and of size $\lambda$. Can we say that ...
7
votes
1answer
242 views

Which of these relations on partial orders allows us to identify forcing equivalence?

Background This question was inspired by Justin Palumbo's excellent question Cantor Bernstein for notions of forcing. In his question, Justin considers a relation $\lhd$ on partial orders (defined ...
8
votes
2answers
516 views

Statements forced by one condition of a poset, but not the whole thing

In order to get the relative consistency of some statement, it suffices to find a notion of forcing, and a condition $p$ in that forcing, such that $p$ forces the desired statement. It seems to be ...
5
votes
1answer
360 views

Does “antichain” mean something different in set-forcing than in lattice theory?

On page 3 of Introduction to Lattices and Order, Davey and Priestley define an antichain in a poset $\langle P,\leq\rangle$ as a set of pairwise incomparable elements: The ordered set P is an ...
7
votes
3answers
580 views

Characterizing forcings that don't add any dominating reals

Regarding reals as functions from $\omega$ to $\omega$, let's say a real $f$ eventually dominates $g$ iff $(\exists n)(\forall m > n)[ f(m) > g(m)]$. Let's say that a (non-trivial separative) ...
2
votes
2answers
502 views

A problem about posets similar to Suslin's problem

Suslin's problem is: Given a complete dense linear order without endpoints, if it has the ccc must it be isomorphic to $\mathbb{R}$? The answer is that it's independent of ZFC. The related ...
7
votes
2answers
377 views

A sequence of generic filters that does not come from an iteration

Fix a countable transitive model $M$ of ZFC. In my answer to this question I indicated that there are forcing iterations $((Q_\alpha:\alpha\leq\omega),(\dot P_\alpha:\alpha<\omega))$ in $M$ and ...