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Topic: 0^0=1
Replies: 28   Last Post: May 8, 2012 8:16 PM

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 Marshall Posts: 1,928 Registered: 8/9/06
Re: 0^0=1
Posted: May 7, 2012 6:00 PM

On Monday, May 7, 2012 10:27:58 AM UTC-7, Jussi Piitulainen wrote:
> Dan Christensen writes:
>

> > So, why not 0^0 = 0? Can this be ruled out using only natural number
> > arithmetic?

>
> Not directly, as far as I can see, but almost nothing supports it
> while much requires x^0 = 1, either for all x or for all natural x.

Using only a very simple definition for the natural numbers,
and adding iterated operators to that, we easily and
naturally come to the conclusion that 0^0 = 1.

Define a natural number as either zero or the successor to
a natural number:

nat := zero | succ nat;

This gives us successor as well, so we can count. Now define
addition as iterated successor. In other words, give a count
for how many times to apply successor:

x + zero := x;
x + succ y := succ x + y;

Now we have addition. Define multiplication as iterated

x * zero := zero;
x * succ y := x + (x * y);

Define iterated multiplication:

x ^ zero := succ zero;
x ^ succ y := x * (x ^ y);

With a little massaging you can put this in your
favorite functional language and execute it directly.
This produces the result that 0^0 = 1.

After I went to the trouble of typing this in I found:

http://www.willamette.edu/~fruehr/LLC/final/lecs/nat/nat5.html

Marshall