Wow, that was a much longer hiatus than I had planned. Let’s see if I can pick this up again.
Last year, I spent several weeks describing in intricate detail what natural numbers are. If I had to describe my thesis in a sentence, this might be it:
“The natural numbers are a computational tool for answering questions about finite sets.”
Here’s a somewhat longer and possibly more helpful recap.
- Sets are ‘pure extensions’: collections of objects that are entirely defined by what is a member, and what is not.
- Given two sets, sometimes there’s a bijective correspondence or bijection between them: a mapping in which every element of the first set is associated to a single element of the second, and vice versa. In this case we say the sets are equipotent, or “the same size”.
- We choose a representative from each class of equipotent finite sets, and call those representatives natural numbers. In this way we can determine whether two finite sets are equipotent just by seeing which natural number each one corresponds to. The natural number associated to a finite set is its cardinality or “size.”
- The ordering and arithmetic on the natural numbers is defined to reflect facts about the sets they describe. For example, addition reflects what happens when two disjoint sets are put together.
An obvious next step would be to give a similar at-length development of rational numbers, integers, real numbers, and so on. But I’d prefer to move in a novel direction, so right now I want you to notice something specific about the construction above, which is how unnecessary was the limitation to finite sets.
There is nothing about the definition of a bijection that says the sets have to be finite! Taking the same construction above, and removing that limitation, won’t create any paradoxes—instead it will give us a whole new collection of numbers to work with. Instead of a computational tool for answering questions about finite sets, we’ll get a tool for answering questions about sets in general.