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Results tagged with forcing
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user 5903
Forcing is a method first used to prove the continuum hypothesis is independent of the classical axioms of set theory
2
votes
Countable support product of Sacks forcings and selective ultrafilters
I'm not allowed to comment yet but your question looks equivalent to having the set $A$ in $\mathrm{HL}_\omega$ (see Laver's paper) belong to the ultrafilter. I do not recall how flexible Laver's proo …
- 11.4k
3
votes
Accepted
Compactifications with remainder $[0,\omega_1]$ and convergent sequences
Here's an example, suggested by Alan Dow. Take a Hausdorff Gap: a pair of sequences $\langle a_\alpha:\alpha<\omega_1\rangle$ and $\langle b_\alpha:\alpha<\omega_1\rangle$ of infinite subsets of $\mat …
- 11.4k
3
votes
$\mathfrak{c}$-universal linear order
You may want to have a look at this paper (PDF) by Baumgartner, Frankiewicz and Zbierski; it establishes the consistency of $\mathrm{MA}_{\sigma\text{-linked}}+\neg\mathrm{CH}$ plus ``every Boolean al …
- 11.4k
5
votes
Difference between Laver's and Mathias's forcing
In this paper Alan Dow compares Laver and Mathias forcing regarding their effects on the algebra $\mathcal{P}(\omega)/\mathit{fin}$. …
- 11.4k
3
votes
Problem understanding a passage of the proof of $\mathfrak{p}=\mathfrak{t}$ involving forcing
Fremlin has the fully correct
$$
D \Vdash \check D\in\dot{\mathcal{G}}
$$
Also, Fremlin did not fix one generic $G$ at the outset; he works with names and the forcing relation everywhere. …
- 11.4k
7
votes
Iterated forcing and CH
Here is a sketch.
We may assume that each $\dot Q_\alpha$ has $\omega_1$ as its universe; in which case the underlying set of $P_{\alpha+1}$ can be taken to be $P_\alpha\times\omega_1$ and the orderin …
- 11.4k
4
votes
Accepted
Copy of $P(\omega)/\mathrm{fin}$ on $\omega_1$
To expand my comment into an answer: take, for each $n\in\omega$, a uniform ultrafilter $u_n$ on $\omega_1$ that contains the set $\{\lambda+n:\lambda$ is a limit or $0\}$.
The set $U=\{u_n:n\in\omega …
- 11.4k