Math Infinity

Conceiving of ∞ as a mathematician is simple. You start counting, and don’t stop.

That’s all.

successor function
($i++ for programmers)

Which is why seems very small to the mind of a mathematician.

With projective geometry you can map to a circle, in which case there is a point-sized hole at the top where you can put ∞ (or −∞, or both).

Same thing with the Riemann Sphere.

stereographic projection of the Riemann sphere

So to them ∞ is very reachable. It’s just a tiny point.

Graham’s Number

It takes much more mental effort to conceive of Graham’s Number than ∞. It took me several hours just to begin to conceive Graham’s number the first time I tried.

underbrace{     {{{{{{{{{{{{3^3}^3}^3}^3}^3}^3}^3}^3}^3}^3}^3}^{cdots}  }       }_{   3^{3^{3^{3^{cdots}}}}  text{ times}  }

Graham’s Number is basically a continuation of the above, recursed many times. Maybe I’ll do a write-up another time but really you can just look at Wikipedia or Mathworld. It’s absolutely mind-blowing.


Here’s what’s weird. Infinity is obviously bigger than Graham’s Number. But Graham’s Number takes up more mental space. Weird, right?

EDIT: Maybe ∞ takes up less mental space than g64 because its minimal algorithmic description is shorter.


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