Tag Archives: Georg Cantor

November 26, 2019

BB #64: Systems Bubble

For the last two weeks I’ve written a number of posts contrasting physical systems with numeric systems.

(The latter are, of course, also physical, but see many previous posts for details on significant differences. Essentially, the latter involve largely arbitrary maps between real world magnitude values and internal numeric representations of those values.)

I’ve focused on the nature of causality in those two kinds of systems, but part of the program is about clearly distinguishing the two in response to views that conflate them.

20 Comments | tags: Alan Turing, Cantor's Diagonal, Georg Cantor, Halting Problem, Heisenberg Uncertainty, Incompleteness Theorems, Kurt Gödel, quantum mechanics, Turing Halting Problem, Werner Heisenberg | posted in Brain Bubble, Computers

June 24, 2014

Sideband #54: Cantor’s Diagonal

By Wyrd Smythe

Be warned: these next Sideband posts are about Mathematics! Worse, they’re about the Theory of Mathematics!! But consider sticking around, at least for this one. It fulfills a promise I made in the Infinity is Funny post about how Georg Cantor proved there are (at least) two kinds of infinity: countable and uncountable. It also connects with the Smooth or Bumpy post, which considered differences between the discrete and the continuous.

This first one is pretty easy. The actual math involved is trivial, and I think it’s fascinating how the Yin/Yang of separate units versus a smooth continuum seems a fundamental aspect of reality. We can look around to see many places characterized by “bumpy” or “smooth” (including Star Trek). (The division lies at the heart of the conflict between Einstein’s Relativity and quantum physics.)

So, let’s consider Cantor.

10 Comments | tags: Cantor, Cantor's Diagonal, finite, Georg Cantor, infinity, integers, irrational numbers, natural numbers, numbers, rational numbers, real numbers | posted in Math, Sideband

October 20, 2012

Infinity is Funny

By Wyrd Smythe

You probably have some idea of what infinity means. Something that is infinite goes on forever. But it might surprise you to know that there are different kinds of infinity, and some are bigger than others!

As an example, a circle has an infinite number of points. (Yet the circumference of the circle has finite length.) Compare that to a straight line with infinite length. Both have infinitely many points but does the finite length circle have fewer? [The answer is no.]

To understand all this, we have to first talk a bit about numbers.

32 Comments | tags: countable, counting numbers, Georg Cantor, infinity, irrational numbers, natural numbers, pi, rational numbers, uncountable | posted in Math, Science