Date: Oct 17, 2013 9:40 AM
Author: Dan Christensen
Subject: Re: Formal proof of the ambiguity of 0^0
On Thursday, October 17, 2013 7:13:01 AM UTC-4, Ben Bacarisse wrote:

> Dan Christensen <Dan_Christensen@sympatico.ca> writes:

>

> <snip>

>

> > There exists infinitely many "exponent-like" functions ? one for each

>

> > natural number x0.

>

> <snip>

>

>

>

> > THEOREM 4: The Product of Powers Rule

>

> >

>

> > ALL(a):ALL(b):ALL(c):[a e n & b e n & c e n

>

> > => [~a=0 => a^b*a^c=a^(b+c)]]

>

> <snip>

>

>

>

> > THEOREM 5: The Power of a Power Rule

>

> >

>

> > ALL(a):ALL(b):ALL(c):[a e n & b e n & c e n

>

> > => [~a=0 => (a^b)^c=a^(b*c)]]

>

> <snip>

>

>

>

> > THEOREM 6 (Lemma): Non-Zero Powers

>

> >

>

> > ALL(a):ALL(b):[a e n & b e n

>

> > => [~a=0 => ~a^b=0]]

>

> <snip>

>

>

>

> > THEOREM 7: The Power of a Product Rule

>

> >

>

> > ALL(a):ALL(b):ALL(c):[a e n & b e n & c e n

>

> > => [~a=0 & ~b=0 => (a*b)^c=a^c*b^c]]

>

>

>

> Have you tried to do the same but with, say, 3^7=14 (leaving 0^0=1)?

>

I'm not sure what you are getting at, but, using my definition, no matter what value you assign to 0^0, you will not get 3^7 = 14. You will only ever get 2,187.

In other words, if you fill in a table values for ^ on N, the value assigned to 0^0 has no effect on the rest of the table. 3^2 will always be 9, etc.

> All the rules simply have the exceptions excluded, so you could do that

>

> with any exception, could you not?

>

To what exceptions are you referring?

Dan

Download my DC Proof 2.0 software at http://www.dcproof.com

Visit my new math blog at http://www.dcproof.wordpress.com