*all numbers with a finite amount of digits *all numbers in finite sets Imagine a set of all whole numbers {0, 1, 2, 3, 4, 5...} And then a set of all integers {...-3, -2, -1, 0, 1, 2, 3...} We can map integers to whole numbers by alternating the signs. {0, -0, 1, -1, 2, -2, 3, -3...} (I'll explain why there's a negative zero in a moment.) Now imagine creating a set of all decimals. Start with 0. Then add 0.1 to 0.9. Then add 0.01 to 0.99, excluding increments of 0.1. Then 0.001 to 0.999 excluding tenths and hundredths, and so on. The set should look like this: {0, 0.1, 0.2, 0.3, 0.4 ... 0.8, 0.9, 0.01, 0.02...} Eventually we can get every 4 digit decimal, 5 digit, 6 digit up to infinity.
The set of all integers (mapped) vs the set of all decimals 0-1 (mapped) can be combined in a table. To further map all this data, we go in a right triangle. Numbers in the grid are denoted by item number. 0 and negative 0 as integers are the same, so we can exclude the 1st item. .0 .1 .2 .3 ... 0 1 3 6 -0 2 4 7 1 5 8 -1 9 ... With an infinite table, we can theoretically get all possible numbers. But never repeating decimals like 0.333333... So this is how we can sort "all" the numbers. Press space to sort the final list. Credit to Garbo for the bubble sort algorithm.
