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Topic: The ambiguity of 0^0 on N
Replies: 106   Last Post: Sep 29, 2013 10:06 AM

 Messages: [ Previous | Next ]
 Dan Christensen Posts: 8,219 Registered: 7/9/08
Re: The ambiguity of 0^0 on N
Posted: Sep 19, 2013 10:19 PM

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

>
>
> 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".

Date Subject Author
9/18/13 Dan Christensen
9/18/13 Peter Percival
9/18/13 Dan Christensen
9/18/13 Peter Percival
9/18/13 Virgil
9/18/13 Dan Christensen
9/18/13 Rotwang
9/18/13 Rock Brentwood
9/18/13 Rotwang
9/19/13 Dan Christensen
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 Dan Christensen
9/20/13 fom
9/19/13 Virgil
9/19/13 Virgil
9/19/13 Rotwang
9/18/13 Virgil
9/18/13 fom
9/18/13 Rotwang
9/28/13 Shmuel (Seymour J.) Metz
9/29/13 Marshall
9/19/13 Dan Christensen
9/19/13 Dan Christensen
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 Michael F. Stemper
9/19/13 Dan Christensen
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 Dan Christensen
9/20/13 fom
9/20/13 Dan Christensen
9/20/13 fom
9/19/13 fom
9/19/13 fom
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 Rotwang
9/19/13 Dan Christensen
9/19/13 Helmut Richter
9/19/13 Dan Christensen
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 fom
9/19/13 JT
9/19/13 JT
9/19/13 Michael F. Stemper
9/19/13 JT
9/19/13 JT
9/19/13 JT
9/19/13 Helmut Richter
9/28/13 Shmuel (Seymour J.) Metz
9/19/13 fom
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 Peter Percival
9/19/13 Karl-Olav Nyberg
9/19/13 fom
9/19/13 fom
9/19/13 Rotwang
9/19/13 Dan Christensen
9/19/13 fom
9/25/13 Rotwang
9/26/13 Dan Christensen
9/27/13 Brian Q. Hutchings
9/19/13 fom
9/18/13 Rock Brentwood
9/19/13 Dan Christensen
9/19/13 Dan Christensen
9/19/13 Rotwang
9/19/13 Dan Christensen
9/19/13 fom
9/20/13 Dan Christensen
9/20/13 fom
9/20/13 Dan Christensen
9/20/13 Peter Percival
9/20/13 Peter Percival
9/20/13 Dan Christensen
9/20/13 Virgil
9/20/13 Peter Percival
9/20/13 fom
9/20/13 Michael F. Stemper
9/20/13 LudovicoVan
9/21/13 Michael F. Stemper
9/21/13 LudovicoVan
9/21/13 Richard Tobin
9/20/13 Peter Percival
9/20/13 Peter Percival
9/21/13 Dan Christensen
9/19/13 Karl-Olav Nyberg