Date: Sep 19, 2013 10:19 PM
Author: Dan Christensen
Subject: Re: The ambiguity of 0^0 on N

On Thursday, September 19, 2013 7:36:04 PM UTC-4, fom wrote:
> On 9/19/2013 1:26 PM, Dan Christensen wrote:
>

> >>
>
> >> Why apart from? Why are you leaving it out?
>
> >
>
> > We can't divide by 0. Unless you want to assign a value to 0/0 as well.
>
> >
>
>
>
> And, what about division is "number theoretic"?
>


As you can see in my proof, I am actually using the right-cancelability property of natural number multiplication:

x*y = z*y & y=/=0 => x=z

"Dividing" both sides by y, to cancel off the factor of y, OK?

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





>
>
> What axiom of number theory ensures closure under division?
>
>
>
> You cannot distinguish an additive group from a multiplicative
>
> group except in relation to a ring structure. Otherwise
>
> the difference is mere purport.
>
>
>
> Relative to ring theory, one may speak of division as some
>
> essential numeric operation. In fact, that is what characterizes
>
> "division rings".
>
>
>
> http://en.wikipedia.org/wiki/Division_ring
>
>
>
> So, you reject "set theoretic" considerations, but you invoke
>
> "ring theoretic" considerations (or, since I am not going to
>
> research obscure notions of "division", some other
>
> "???-theoretic" considerations which are not "number-theoretic"
>
> considerations).
>
>
>
> You cannot invoke "number theory" as a restriction and then
>
> use arguments outside of "number theory" in your defense.
>
>
>
> In "number theory" you cannot divide by 2 unless you wish
>
> to assign natural numbers to
>
>
>
> 1/2, 3/2, 5/2, ..., (2n+1)/2, ...
>
>
>
> n in {0, 1, 2, ...}
>
>
>
>
>
> The standard definitions based upon the established operations
>
> of the language of the theory suffice to prove the fundamental
>
> theorem. If you wish to begin from the exponentiation associated
>
> with the fundamental theorem then you need to present a different
>
> theory for consideration as "number theory".