On Wednesday, September 18, 2013 8:33:15 PM UTC-4, Rotwang wrote:
> > The problem is that Dan has things backwards; rather than picking a > > definition and determining what properties it satisfies, he's picked a > > few properties satisfied by the usual definition, noticed that those > > properties aren't sufficient to derive the usual definition, and is now > > insisting that we should therefore abandon the usual definition unless > > someone can show that the other possible definition that satisfies those > > properties gives rise to a contradiction. This makes no sense.
Some elements of what you call the "usual definition" are redundant. They can actually be derived from other elements. (See my proof.)
Another element of your "usual definition" -- your 0^0=1 -- may even lead to a contradiction. I guess that's why they leave it out in other "usual definitions."