Tag Archives: Set Theory

Category Theory, Isomorphism, Functor (More Hiking in Modern Math World 7/7)

March 06, 2016More Hiking in Modern Math World2284 vuesEdit

Infinity, Set Theory, Continuum Hypothesis, Incompleteness (Hiking in Modern Math 3/7)

February 16, 2016More Hiking in Modern Math World1069 vuesEdit

Set Theory, Russell, Gödel (Trek through Math 6/8)

February 05, 2016A Trek through 20th Century Mathematics1125 vuesEdit

The Limitless Vertigo of Cantor’s Infinite

January 29, 2015Article2163 vuesEdit
No one believed him. Not even fellow mathematicians. They thought he was wrong. They thought he was crazy. Even he ended up doubting himself and went crazy. And yet, he had mathematically proved it all. Georg Cantor had figured out how to manipulate the infinite. Even more remarkable, he showed that there were actually several infinities; and some are bigger than others!

Univalent Foundations of Mathematics

April 21, 2014Article2366 vuesEdit
In an effort to make mathematics more computable, a consortium of today's greatest mathematicians have laid out new foundations. Amazingly, they all lie upon one single axiom, called univalence. The goal of this axiom is to make formal mathematics more similar to informal mathematics. With univalence, our Arabic numbers aren't just like natural numbers; They are natural numbers. Univalence also has unforeseen and mesmerizing consequences.

Homotopy Type Theory and Higher Inductive Types

April 06, 2014Article2057 vuesEdit
In this article, we explore the possibilities allowed by higher inductive types. They enable a much more intuitive formalization of integers and new mind-blowing definitions of the (homotopical) circle and sphere.

Type Theory: A Modern Computable Paradigm for Math

March 20, 2014Article8359 vuesEdit
In 2013, three dozens of today's brightest minds have just laid out new foundation of mathematics after a year of collective effort. This new paradigm better fits both informal and computationally-checkable mathematics. There is little doubt that it will fundamentally change our perspective on rigorous knowledge, and it could be that, in a few decades, the book they published turns out to be the bedrock of all mathematics, and, by extension, all human knowledge! Have a primer of this upcoming revolution, with this article on type theory, the theory that the book builds upon!

Construction and Definition of Numbers

December 07, 2012Article4655 vuesEdit
Although they have been used for thousands of years, an actual definition of numbers was given less than a century ago! From the most fundamental level of set theory, this article takes you to the journey of the construction of natural, integer, rational, real and complex numbers.

Self-Reference, Math Foundations and Gödel’s Incompleteness

November 29, 2012Article3357 vuesEdit
Although highly appreciated by artists, self-reference has been a nightmare for mathematicians. It took one of the greatest, Kurt Gödel, to provide a better understanding of it. This article deals with paradoxes, recursion, fractals and Gödel's incompleteness theorems.