gloftus

Countability is the whole point, there's no need to apologize. I was merely offering the perspective that "towers of infinity" is possibly the least useful consequence that comes from defining the notion of countability. To my mind, what we really reap from Cantor's work is a better understanding of the topology of the real numbers. But you have to define countability first in order to understand what uncountability really implies.

Worth noting that the hyperbolic triangle in the article contains "points at infinity" which are not actually a part of the hyperbolic plane, so this is really an "improper triangle" as the article notes. One could construct a similar improper triangle in the Euclidean plane that consisted of two parallel lines meeting at infinity. Such a triangle would still have 180 degrees of internal angle but it's area and perimeter would be infinite.

Ego death.

You can't enumerate the real numbers, but you can grab them all in one go - just draw a line!

The more I learn about this stuff, the more I come to understand how the quantitative difference between cardinalities is a red herring (e.g. CH independent from ZFC). It's the qualitative difference between these two sets that matter. The real numbers are richer, denser, smoother, etc. than the natural numbers, and those are the qualities we care about.

This isn't iconoclasm, it's ignorance.

Yes, graphs are ubiquitous because they are so abstract. They live on the same level of abstraction as pure numbers. There are useful "numerical" libraries that exist, and by analogy I think you could say there are also useful "graphical" libraries that exist. But we don't really have "number" libraries, and we don't really have "graph" libraries, because those concepts are a bit too abstract to write APIs against.

I think you are being down voted because you appear to have an ax to grind rather than a useful perspective on the issue.