5 divided by 0? Personally, I believe that it is infinite based on the idea that division is just repeated subtraction just like multiplication is repeated addition. For example, in 4/2, it's pretty much like saying how many times can you subtract 2 from 4 before you get to 0.
I would give a slightly different moral to Peter's story. Mathematicians could have defined 5 divided by 0 to be infinity--one of the wonderful things about mathematics is that we can define things however we want. However, what Peter's proof shows is that if you define division by 0, then some of the familiar algebraic laws aren't going to work anymore. (It is an interesting exercise to identify the algebraic law used in the proof that would stop working if we defined division by 0.) So it would actually be quite inconvenient to change the usual definition of division, according to which division by 0 is undefined.