Infinite Number of Infinities

When I was a kid, I would get in arguments with my siblings where we’d say things like, “you’re dumb times infinity!” “Well, you’re dumb times infinity squared!” And the clincher was, “you’re dumb times infinity to the infinity power!” First of all, turns out that “infinity,” “infinity squared” and “infinity to the infinity power” are all the same cardinal number. Meaning that there is exactly the same amount in “infinity to the infinity power” as there is in just plain old “infinity.” This is even true for “infinity to the infinity power to the infinity power to the…” infinitely many times. All the same number.

However, there is a larger brand of infinity that makes this infinity look pretty dinky. If you take all of the decimal numbers between 0 and 1 (e.g. .017489182…. and .167588701….) then there is a greater quantity of these numbers than “infinity to the infinity power.” And it turns out that there is another infinity even bigger than this infinity. In fact, there are infinitely many infinities out there, just waiting for us to blow our minds trying to think about them.

More about this one at a later time…

ps: Here's the statement of the famous "continuum hypothesis" which I have no reason not believe (look it up, it's an interesting one):