Take the set of positive integers (0, 1, 2…). Double them, so you have (0, 2, 4…). You had an infinite list of numbers, all "compact" (no room for more positive

Take the set of positive integers (0, 1, 2…). Double them, so you have (0, 2, 4…). You had an infinite list of numbers, all "compact" (no room for more positive integers inbetween them), and with a simple mathematical transform, you've given yourself space for a second equally-sized infinity of integers to fit neatly inside of them.