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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 12:04 PM

Michael F. Stemper wrote:
> 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.

I agree in all respects! Betcha Dan doesn't!

--
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