Long iterations for the continuum

Countable support iterations and large continuum

We prove that any countable support iteration formed with posets with $ω_2$-p.i.c.\ has $ω_2$-c.c., assuming CH in the ground model and assuming also that $ω_1$ is not collapsed. This improves earlier results of Shelah by removing the restriction on the length of the iteration. Thus, we solve the problem of obtaining a large continuum via such forcing iterations.

一个与大连续统相容的通用绝对性原理

A generic absoluteness principle consistent with large continuum

本研究提出一种新的通用绝对性原理，并借助Shelah记忆迭代技术验证了该原理与大连续统的相容性，为数理逻辑相关研究提供了新的参考。

Forcing consequences of PFA together with the continuum large

We develop a new method for building forcing iterations with symmetric systems of structures as side conditions. Using our method we prove that the forcing axiom for the class of all the small finitely proper posets is compatible with a large continuum.

Measuring club sequences, together with the Continuum Hypothesis

We answer a question of Moore by building a forcing extension satisfying measuring together with CH. The construction works over any model of ZFC and can be described as a forcing iteration with countable structures as side conditions and with symmetry constraints. Also, we show that a small variation of this construction produces a model of measuring together with the continuum being larger than the second uncountable cardinal.

Large continuum, oracles

Our main theorem is about iterated forcing for making the continuum larger than aleph_2. We present a generalization of math.LO/0303294 which is dealing with oracles for random, etc., replacing aleph_1, aleph_2 by lambda,lambda^+ (starting with lambda=lambda^{<lambda}>aleph_1). Well, instead of properness we demand absolute c.c.c. So we get, e.g. the continuum is lambda^+ but we can get cov(meagre)=lambda. We give some applications. As in math.LO/0303294, it is a "partial" countable support iteration but it is c.c.c.

Template iterations with non-definable ccc forcing notions

We present a version with non-definable forcing notions of Shelah's theory of iterated forcing along a template. Our main result, as an application, is that, if $κ$ is a measurable cardinal and $θ<κ<μ<λ$ are uncountable regular cardinals, then there is a ccc poset forcing $\mathfrak{s}=θ<\mathfrak{b}=μ<\mathfrak{a}=λ$. Another application is to get models with large continuum where the groupwise-density number $\mathfrak{g}$ assumes an arbitrary regular value.