At this rate, I will regurgitate everything I’ve ever learned, and advance the timeline farther than I have years to claim. I guess it could be worse. Here is the construction of a number system.

First we have elements, and we select them gradually into a growing group: one in one, representing any such class of one elements, and designated with the cardinal number as its size “1.” Then we add a second element and the group grows, and this we represent with the digit-symbol “2.” And we go forever.

Three observations:

- Each cardinal number is finite. By induction, first 1: there is no part of one element that would be its equivalent. The next (2) is finite because adding one element to a finite group – the group containing one element – does not disturb the property of the new group being finite [1]. So the group II is finite because I is, and III is finite because II is, etc.
- All cardinals are in this list. We can take a group and mark each element within it, and the last mark will be finite. That would be the cardinal number. If our counting never ended, the group would be equivalent to a part of itself, and therefore would be infinite.
- No numbers are equal to each other. Any two finite groups are either equivalent or one is less than the second. Our construction assures us the latter, that each group is a part of its subsequent.