On 19/09/2013 06:41, Dan Christensen wrote: > 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.)
What I call the usual definition is this: ^ is the unique function with domain NxN that satisfies, for all n, m in N,
n^0 = 1 n^(m + 1) = n*n^m
Please show me which elements of that are redundant.
> Another element of your "usual definition" -- your 0^0=1 -- may even lead to a contradiction.