Date: Nov 1, 2013 9:38 AM
Author: Dan Christensen
Subject: Re: Formal proof of the ambiguity of 0^0

On Friday, November 1, 2013 6:05:18 AM UTC-4, Bart Goddard wrote:
> Dan Christensen <Dan_Christensen@sympatico.ca> wrote in
>
> news:727a4503-3f63-4528-855c-fb30c7d24d8d@googlegroups.com:
>
>
>

> > Here, I prove 0^x = 0 for x=/=0
>
>
>
> The proof is fatally flawed. If y = 1, then z = 0,
>
> in which case the "proof" is threaded with the
>
> "uspecified" phrase "0^0."


On the contrary, this is not a problem at all. Again, 0^0 is a natural number, but no value is assigned to it in my definition of ^ on N. This formalizes the longstanding practice (since Cauchy in the early 19th century) of leaving 0^0 undefined. Again, I define ^ as follows:

1. ALL(a):ALL(b):[a e n & b e n => a^b e n] (i.e. ^ is a binary function on N)

2. ALL(a):[a e n => [~a=0 => a^0=1]]

3. ALL(a):ALL(b):[a e n & b e n => a^(b+1)=a^b*a]

These 3 statements characterize ALL exponent-like functions on N (Theorem 1). And ALL exponent-like functions on N are identical except for the value assigned to (0,0). (Theorem 2).

No value is assigned to 0^0, but as readers can see for themselves from the theorems posting here, it is really quite easy to work around this minor inconvenience.

Dan
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