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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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 Michael F. Stemper Posts: 125 Registered: 9/5/13
Re: Leaving 0^0 undefined -- A number-theoretic rationale
Posted: Sep 12, 2013 11:45 AM

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.

--
Michael F. Stemper
Always use apostrophe's and "quotation marks" properly.