On Thursday, September 19, 2013 12:50:02 PM UTC-4, Rotwang wrote: > On 18/09/2013 15:53, Dan Christensen wrote: > > > For the natural numbers, exponents greater than 1 are naturally defined for x in N as follows: > > > > > > x^2=xx > > > x^3=xxx > > > x^4=xxxx > > > x^5=xxxxx > > > > > > and so on. > > > > > > More formally, exponentiation can be defined as a binary function on the set of natural number N such that: > > > > > > (1) Ax in N: x^2=x*x > > > > Let's see your proof of that identity. Sorry, simply defining it as such > > won't do in this context.
It is not a theorem that requires a proof. It is a definition, or if you like, an axiom. Unlike your 0^0=1, it is justified by the natural development of exponentiation that I give above.