Flow: the Axiom of Choice is independent from the Partition Principle in ZFU

ZFU 中的反射原理

Reflection Principles in ZFU

ZFU中分离了集合原理、反射原理和部分反射原理，并探讨了它们在不同公理假设下的等价性与独立性。

含本原元素集合论的公理化与力迫

Axiomatization and Forcing in Set Theory with Urelements

本文系统研究含本原元素集合论中的公理层级，并提出新的力迫名称定义以证明力迫基本定理，进而揭示力迫对公理的破坏与恢复以及地面模型可定义性的失效现象。

Flow: the Axiom of Choice is independent from the Partition Principle

We introduce a general theory of functions called Flow. We prove ZF, non-well founded ZF and ZFC can be immersed within Flow as a natural consequence from our framework. The existence of strongly inaccessible cardinals is entailed from our axioms. And our first important application is the introduction of a model of Zermelo-Fraenkel set theory where the Partition Principle (PP) holds but not the Axiom of Choice (AC). So, Flow allows us to answer to the oldest open problem in set theory: if PP entails AC.

Ramsey theory over partitions I: Positive Ramsey relations from forcing axioms

In this series of papers, we advance Ramsey theory of colorings over partitions. In this part, a correspondence between anti-Ramsey properties of partitions and chain conditions of the natural forcing notions that homogenize colorings over them is uncovered. At the level of the first uncountable cardinal this gives rise to a duality theorem under Martin's Axiom: a function $p:[ω_1]^2\rightarrowω$ witnesses a weak negative Ramsey relation when $p$ plays the role of a coloring if and only if a positive Ramsey relation holds over $p$ when $p$ plays the role of a partition. The consistency of posi…

组合二分性与基数不变量

Combinatorial dichotomies and cardinal invariants

假设的P-理想二分法，我们试图隔离的连续体，这是与适当的强制公理的两个众所周知的后果相关的主要特征。我们发现了一个基数不变的$\mathfrak{x}$，使得$\mathfrak{x} > ω_{1}$等价于1，$ω$，$ω_{1}$，$ω\times ω_{1}$和${\left[ω_{1}\right]}^{< ω}$是唯一的大小至多为${\aleph}_{1}$的有向集的共尾类型。我们调查相应的问题的划分关系$ω_{1} \rightarrow (ω_{1}, α)^2$的所有$α< ω_{1}$。为此，我们调查的分区关系的一个连贯的Suslin树$\mathbb{S}$的可比元素对。我们表明，一个积极的分区关系，这样的对如下从最大量的适当的强制公理兼容的存在$\mathbb{S}$。因此，我们得出结论，在强迫与连贯的Suslin树$\mathbb{S}$地面模型满足这种相对化的适当的强制公理，$ω_{1} ~\rightarrow {(ω_{1}, α)}^{2}$的所有$α< ω_{1}$。我们证明，这种积极的分区关系，$\mathbb{S}$不能得到改善，在$\mathrm{ZFC}$显示，$\mathbb{S} \not\rightarrow ({\aleph}_{1}, ω+2)^2$。

An Introduction to Combinatorics of Determinacy

This article is an introduction to combinatorics under the axiom of determinacy with a focus on partition properties and infinity Borel codes.