```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? DanDownload 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".
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