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.
However, by the fith axiom of euclid, the lines in your example cannot be parallel AND converge (not even in infinity). Thus, it's more an open rectangle.
Either they are overlapping which violates the definition of a triangle, or they don't and the parallel lines always maintain the same distance X to each other and consequently maintain distance X at infinity (let's say X=1, bc you can just scale it).
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.
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.
I'm aware that I'm ignorant of many things, just like anyone else on this Earth. Some are less ignorant and some are more.
Could you be kind enough to explain the phrase "set of all integers" when the word all can not apply to an unbounded quantity? I think the word all is used loosely to extend it's meaning as used for finite sets, to a non-existent, unbounded set. For example, things such as all Americans, all particles in universe have a meaning because they have a boundary criterion. What is all integers?
I think one need to first define the realm of applicability or domain for the concepts such as comparison, 1-to-1 mapping, listing, diagonals, uniqueness, all etc.
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.
it's true that numbers are very abstract, which is what makes it so easy to design apis for them
the python runtime includes four built-in number types (small integer, arbitrary-precision integer, float, and complex) and the python standard library includes two more number types (decimal and fractions), and one of the most popular non-standard libraries for python is numpy, which provides some other kinds of numbers such as single-precision floats, vectors, and matrices. other systems like pari/gp have number libraries that provide other kinds of numbers, such as p-adic numbers and galois field elements
the only programming languages i've ever used that didn't have 'number' libraries were esoteric languages like brainfuck and the lambda calculus
numbers have all of mathematics as a background which is what makes it so easy to design apis for them
graphs are a much newer development, I think there's a very deep connection between category theory and graphs in general (and also computers make both much more useful somehow)
lambda calculus can be used to define numbers but it's a wonky construction, it's reminiscent of how sets can also be used to define numbers.
