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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 11:25 PM

On Thursday, September 19, 2013 11:07:23 PM UTC-4, fom wrote:
> On 9/19/2013 9:19 PM, Dan Christensen wrote:
>

> > 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?
>
> >
>
>
>
> Not really.
>
>
>
> Cancellability is a feature of group operations or, more
>
> generally magmas.
>

It is also a feature (or theorem) of natural number arithmetic, e.g. x*2 = y*2 => x=y

>
>
> http://en.wikipedia.org/wiki/Cancellation_property
>
>
>
> So, again, you are invoking a property that is not strictly
>
> "number theoretic".
>

I disagree.

>
>
> The axiom of Peano's number theory that addresses equality
>
> between natural numbers (other than transitivity, symmetry,
>
> reflexiveness, and closure) is given by
>
>
>
> x=y <-> ( x+1 ) = ( y+1 )
>
>
>
> Wikipedia lists this as a conditional,
>
>
>
> ( x+1 ) = ( y+1 ) -> x=y
>
>
>
> It is axiom 8,
>
>
>
> http://en.wikipedia.org/wiki/Peano_axioms#The_axioms
>
>
>
> Now, since you continue to invoke "number theory", perhaps
>
> you could provide the formal theory which you seem to
>
> think that everyone knows and loves. It is clearly not
>
> Peano arithmetic.

If you won't accept x*2 = y*2 => x=y, I afraid there is not much point in continuing this discussion.

Dan

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