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Topic: Leaving 0^0 undefined -- A number-theoretic rationale
Replies: 48   Last Post: Sep 15, 2013 1:06 PM

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 Peter Percival Posts: 2,623 Registered: 10/25/10
Re: Leaving 0^0 undefined -- A number-theoretic rationale
Posted: Sep 12, 2013 3:50 AM

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.
>>

>
> You are talking about some other set-theoretic notion of exponentiation, not the usual arithmetic operator that we are discussing here.
>
> More convincing would be obtaining a contradiction by assuming only 0^0=0 along with the usual rules of natural-number arithmetic, including the usual Laws of Exponents.

On the natural numbers ^ is defined by

x^0 = 1
x^{y+1} = x*x^y.

Look at the definitions of + and * in PA and you'll see it's a common theme.

--
Sorrow in all lands, and grievous omens.
Great anger in the dragon of the hills,
And silent now the earth's green oracles
That will not speak again of innocence.
David Sutton -- Geomancies