Gödel’s First Incompleteness Theorem for Programmers

Gödel’s incompleteness theorems have been hailed as “the greatest mathematical discoveries of the 20th century” — indeed, the theorems apply not only to mathematics, but all formal systems and have deep implications for science, logic, computer science, philosophy, and so on. In this post, I’ll give a simple but rigorous sketch of Gödel’s First Incompleteness … [Read more…]

Confusing Math with Morality

This essay is in response to Counterintuitive problem: Everyone in a room keeps giving dollars to random others. You’ll never guess what happens next. Dan Goldstein attributes the problem to Uri Wilensky, of Northwestern, who formulates it thusly: Imagine a room full of 100 people with 100 dollars each. With every tick of the clock, … [Read more…]

How to prove there is no Universe

Years ago, when I first read Paul Halmos’ seminal Naive Set Theory, I was blown away by how easy it was to prove that there is no universe. In fact, not even three sections in, he drops this italicized bombshell: $$nothing\text{ }contains\text{ }everything$$ or, “more spectacularly,” he continues $$there\text{ }is\text{ }no\text{ }universe$$ Luckily, we only … [Read more…]