On 09/12/2013 02:46 AM, Peter Percival wrote: > Dan Christensen wrote: >> On Wednesday, September 11, 2013 5:09:52 PM UTC-4, Peter Percival wrote: >>> Dan Christensen wrote:
>>>> Show me a contradiction that arises from 0^0 = 0.
>>> The product of the empty set is 1, hence 0^0 = 1. That contradicts 0^0 >>> = 0. Therefore 0^0 doesn't = 0.
>> More convincing would be obtaining a contraction by assuming only >> 0^0=0 along with the usual rules of natural-number arithmetic, >> including the Laws of Exponents. THAT would be interesting. > > Among the usual rules is 0^0=1.
I'm not real good at boundary cases, but wouldn't the empty set be a function from the empty set to the empty set? If so, then this doesn't even need to be a special rule -- it's derivable from the definition of exponentiation in cardinal arithmetic. Since everything else in number theory is equivalent to cardinal arithmetic, it'd be pretty strange to create a special rule to throw this out.
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