Mathematical examples

This is very simple. There's one counterexample to the diagonal argument, the points of the line drawn from zero to one.

Correspondingly, the density of the rationals in the reals puts their intervals 1-1 with those of other sets dense in the reals (as they are also so dense).

These plain number-theoretic facts are neatly derivative of first principles, and the conscientious mathematician is now interested in how that is.