More on: the revised GCH and middle diamond

Shelah修正GCH定理浅探

A short tour of Shelah's Revised GCH Theorem

本文梳理并比较了Shelah原始论文中修正GCH定理的不同表述形式，探讨了各表述的含义。

相遇、覆盖与Shelah的修正广义连续统假设（GCH）

Meeting, covering and Shelah's Revised GCH

本研究重新探讨Shelah修正GCH定理在钻石原则中的应用，提出该定理的推广形式并证明其一小片段，还考察了定理对形如cov(-, -, -, $ω$)的覆盖数的应用。

A note on the Revised GCH

We give a proof of Theorem 2.10 from [8] that eliminates the use of Shelah's nice filters and associated rank functions, and instead uses only the well-foundedness of reduced products of ordinals modulo countably complete filters. This removes any need to make assumptions about the existence of large cardinals in core models.

Essentially disjoint families, conflict free colorings and Shelah's Revised GCH

Using Shelah's revised GCH theorem we prove that if mu<beth_omega <= lambda are cardinals, then every mu-almost disjoint subfamily B of [lambda]^{beth_omega} is essentially disjoint, i.e. for each b from B there is a subset f(b) of b of size < |b| such that the family {b-f(b) b in B} is disjoint. We also show that if mu<=kappa<=lambda, and kappa is infinite, and (x) every mu-almost disjoint subfamily of [lambda]^kappa is essentially disjoint, then (xx) every mu-almost disjoint family B of subsets of lambda with |b|>=kappa for all b from B has a conflict-free colorings with kappa colors. Puttin…

The Generalized Continuum Hypothesis revisited

We argue that we solved Hilbert's first problem positively (after reformulating it just to avoid the known consistency results) and give some applications. Let lambda to the revised power of kappa, denoted lambda^{[kappa]}, be the minimal cardinality of a family of subsets of lambda each of cardinality kappa such that any other subset of lambda of cardinality kappa is included in the union of <kappa members of the family. The main theorem says that almost always this revised power is equal to lambda. Our main result is The Revised GCH Theorem: Assume we fix an uncountable strong limit cardinal…