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

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 Dan Christensen Posts: 8,219 Registered: 7/9/08
Re: The ambiguity of 0^0 on N
Posted: Sep 19, 2013 11:40 AM
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On Thursday, September 19, 2013 11:34:54 AM UTC-4, Michael F. Stemper wrote:
> On 09/19/2013 09:51 AM, Dan Christensen wrote:
>

> > On Thursday, September 19, 2013 7:26:55 AM UTC-4, Peter Percival wrote:
>
> >> Dan Christensen wrote:
>
> >>
>
> >>> On Wednesday, September 18, 2013 7:11:53 PM UTC-4, Rotwang wrote:
>
> >>
>
> >>
>
> >>
>
> >>>>
>
> >>
>
> >>>> will follow from people defining exponentiation in the usual way?
>
> >>
>
> >>>
>
> >>
>
> >>> Whatever consequences may arise from a calculation that results in a
>
> >>
>
> >>> value of 1 when it should be 0. The result could be catastrophic.
>
> >>
>
> >>
>
> >>
>
> >> When "should" 0^0 = 0?
>
> >>
>
> >
>
> > The value of 0^0 should not depend on the context.
>
> >
>
> >>
>
> >>
>
> >> This is what has happened. You were writing a computer program and you
>
> >>
>
> >> wrongly believed that exp(0,0)--or whatever the notation is--would
>
> >>
>
> >> return 0. It didn't, it returned--quite properly--1. Your program
>
> >>
>
> >> being buggy is not a catastrophe. The thing to do with bugs is to fix
>
> >>
>
> >> them[*], _not_ try to "fix" mathematics.
>
> >>
>
> >
>
> > Let's see your proof that 0^0=1. Sorry, simply defining it as such won't do in this context.
>
>
>
>
>
> There is a simple definition of exponentiation on N, based on
>
> cardinal arithmetic. It doesn't have any special cases, but
>
> 0^0 = 1 falls out of it quite naturally:
>
>
>
> |A|^|B| = |{f | f:B->A}|
>
>
>
> In English, the cardinal number of set A raised to the power
>
> of the cardinal number of set B is the cardinal number of the
>
> set of all functions from B to A. No exceptions, no special
>
> cases.
>

Thanks, but as with any analogy, it may not be perfect.

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

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

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