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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: 8,155 Registered: 7/9/08
Re: The ambiguity of 0^0 on N
Posted: Sep 18, 2013 11:32 AM

On Wednesday, September 18, 2013 11:08:01 AM UTC-4, Peter Percival wrote:
> Dan Christensen wrote:
>
>
>

> >
>
> > More formally, exponentiation can be defined as a binary function on
>
> > the set of natural number N such that:
>
>
>
> Can be but isn't, except by you. Everyone else, when defining a
>
> function on N, will start where N starts, viz at 0.
>
>
>

> >
>
> > (1) Ax in N: x^2=x*x
>
> >
>
> > (2) Ax,y in N: x^(y+1) = x^y * x
>
> >
>
> > It can then be shown that:
>
> >
>
> > (1) Ax in N:(x=/=0 => x^1=x)
>
> >
>
> > (2) Ax in N:(x=/=0 => x^0=1)
>
> >
>
> > (3) Ax,y,z in N:(x^(y+z) = x^y * x^z) => 0^1=0 /\ (0^0=0 \/ 0^0=1)
>
> >
>
> > (Formal proof to follow.)
>
> >
>
> > Thus, if the Product of Powers Rule is to hold on N, 0^0 will be
>
> > ambiguous -- being either 0 or 1. Unless one of these alternatives
>
> > can be formally proven
>
>
>
> 0^0 = 1 _can_ be formally proven, if you use the right definition.
>

I think you mean it can be "proven" if you define it as such.

>
>
> If your definition of ^ doesn't tell you what 0^0 is (remember, we're
>
> talking about the naturals here) then the definition is incomplete.
>

Perhaps in the sense that it doesn't define a unique function -- just a function on N with certain properties.

>
>
> If you want to define ^ on the positive integers only and then claim
>
> that 0^0 isn't thereby defined, then you'd be right. But it is _you_
>
> who claim to be talking about N.
>

It should clear from the context that I am talking about the natural numbers including 0.

>
>

> > or shown to give rise to a contradiction, the
>
> > prudent course is to leave 0^0 undefined.
>

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