This question is inspired by a lecture Bjorn Poonen gave at MIT last year. I have ideas of my own, but I'm interested in what other people have to say, so I'll make this community wiki and post my own thoughts later. Here are some examples of what I'm talking about:
- Why does a^0 = 1?
- Why does 0! = 1?
- If the Fibonacci number Fn+1 counts the number of ways to tile a board of length n with tiles of length 1 and 2, why does F1 = 1?
- What is the determinant of a 0x0 matrix?
- What is the degree of the zero polynomial?
- What is the direct product of zero groups?
- What is the zeroth homotopy group of a space?
I want to be very precise about exactly what I'm asking for here.
Question 1: What general principles do you apply in a situation like this? Can they be stated as theorems, or do they only exist at the level of intuition?
Question 2: Do you know of any examples where there are two different ways to extend a sequence to zero, both of which are reasonable from the perspective of some principle?
Feel free to answer at any level of sophistication.