Lu, Yukuan

1. Tree of Thoughts: Deliberate Problem Solving with Large Language Models
2. Self-Consistency Improves Chain of Thought Reasoning in Language Models

1. Paul's Online Notes - Calculus

1. EE 376A: Information Theory. Winter 2018. Lecture 4. - Stanford University
2. EE 376A: Information Theory. Winter 2017. Lecture 4. - Stanford University
3. Elements of Information Theory

Ref:

1. EE 376A: Information Theory. Winter 2018. Lecture 4. - Stanford University
2. EE 376A: Information Theory. Winter 2017. Lecture 4. - Stanford University
3. Elements of Information Theory

Sources:

1. EE 376A: Information Theory. Winter 2018. Lecture 4. - Stanford University
2. EE 376A: Information Theory. Winter 2017. Lecture 4. - Stanford University
3. Elements of Information Theory

结构主义(Constructivism)数学是一种数学哲学. 它折射出的是对于数学本质的思考以及对现有的以公理化集合论为基础的严谨但不适应人类直觉的数学体系的不满.

->Sources: Intuitive mathematics - Marianne Freiberger

This post introduces set theory. The naive theory encountered some paradoxes during 19 century. As a result, people created axiomatic set theories.

Among them, ZFC is the basic axiom system for modern (2000) set theory.

在数学中, 无穷和极限是相当令人困惑的概念.

本文介绍"无穷(infinity)"这一概念, "无穷集合"所具有的反常性质, 以及现代集合公理系统(ZFC)对于无穷集合的处理.

这也引申出measure theory(测度论)这一领域的动机: 在ZFC系统下, 在处理无穷集合时, 传统的基于直觉的测度定义不再适用. 现代的测度理论首先规定"可测集合(measurable sets)"这一概念, 对于可测集合进行其测度的讨论.

最后, 本文给出两个例子, 即大名鼎鼎的Hilbert's paradox of the Grand Hotel和Banach–Tarski Paradox. 以揭示两个事实:

1. 无穷集合具备和有穷集合不同的性质, 有些性质甚至是相当反直觉的.
2. ZFC公理系统(包括选择公理Axiom of choice)会推导出一些反直觉的结论.

Ref: An Introduction to Measure Theory - Terence Tao

在数学中, 有限大小的集合很好处理, 但无限大小的集合却并非如此. 本文探讨集合的可数性, 某些无限大小集合是不可数的, 例如全体实数\(\mathbb R\)和全体有理数\(\mathbb R /\ \mathbb Q\).

Ref:

1. Concepts of Mathematics from CMU.

Operations on iTerm2.