Conceiving of ∞ as a mathematician is simple. You start counting, and don’t stop.
$i++ for programmers)
Which is why ∞ seems very small to the mind of a mathematician.
With projective geometry you can map ℝ 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.
So to them ∞ is very reachable. It’s just a tiny point.
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.
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.