empty promises

I promise that this blog won't always be about the empty set, but ...

Within the category of sets, Ø is what is called an initial object. That just means that for every set S there is exactly one "morphism" from Ø to S, in other words one function which associates to every member of the empty set a member of S. (The word "every" in this last phrase may take some getting used to.)

The empty set is also "initial" in a different sense. The reason why it took Russell and Whitehead hundreds of pages to get their foundations for mathematics developed to the point where they could prove that 1+1=2 was that they were defining "number", and some particular numbers, from scratch, or rather from set theory, and then defining "addition". They thought of the number 2 as, more or less, the set of all sets having exactly two elements; but in order to make sense of that idea in a non-circular fashion they had to actually define 2 to be the set of all sets of the same "size" as some particular set having just two elements. The set that they chose for this purpose was the set whose elements are the numbers 0 and 1. So they had to define 0 and 1 before defining 2. Of course, 1 was the set of sets of the same size as the set consisting of 0 alone. And of course 0 was the set of all sets having the same size as the empty set. Something like that. So the empty set was a starting point.

For me, part of the appeal of emptiness (and now I am not talking particularly about the empty set, but of empty vessels, blank minds, open-ended questions, clear skies... ) is the sense of beginning. So I'm glad that in category theory Ø is called an initial object.

On the other hand, a synonym for initial object is universally repelling object. So much for the found poetry of technical jargon.