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Topic: Equivalent Definition of Exponentiation on N
Replies: 85   Last Post: Dec 2, 2013 10:07 PM

 Messages: [ Previous | Next ]
 Jussi Piitulainen Posts: 355 Registered: 12/12/04
Re: Equivalent Definition of Exponentiation on N
Posted: Nov 26, 2013 1:18 AM

Shmuel (Seymour J.) Metz writes:

> In <qothabdo2zv.fsf@ruuvi.it.helsinki.fi>, on 11/15/2013
> at 09:53 PM, Jussi Piitulainen said:
>

> > Shmuel (Seymour J.) Metz writes:
>
> >> In <l60dev\$144\$1@news.albasani.net>, on 11/13/2013
> >> at 05:38 PM, Peter Percival said:
> >>

> >> > Natural number arithmetic is also up to the job of defining
> >> > 0^0. And clearly it should be 1.

> >>
> >> That depends on what properties you want it to have. If you want
> >> x^{y+1} to be x*x^{y+1}, then you have a problem with 0^0.

>
> > What problem?
>
> That "Ax,y(x^{y+1} >= x^y)" is no longer a theorem.

Did you really have this in mind when you wrote the above? It would
have been much clearer to say "if you want x^{y+1} >= x^y".

But that's also not a theorem when 0^0 is left undefined. The weaker
"Ax,y(x > 0 => x^{y+1} >= x^y)" is a theorem in naturals whether 0^0
is defined or not.

Date Subject Author
11/13/13 Dan Christensen
11/13/13 William Elliot
11/13/13 Dan Christensen
11/15/13 Shmuel (Seymour J.) Metz
11/13/13 Robin Chapman
11/13/13 Peter Percival
11/13/13 Dan Christensen
11/13/13 Peter Percival
11/13/13 Dan Christensen
11/13/13 Bart Goddard
11/13/13 Dan Christensen
11/13/13 Bart Goddard
11/13/13 Dan Christensen
11/13/13 Bart Goddard
11/13/13 Dan Christensen
11/13/13 Bart Goddard
11/13/13 Dan Christensen
11/13/13 Bart Goddard
11/13/13 Dan Christensen
11/13/13 Bart Goddard
11/13/13 Dan Christensen
11/14/13 Peter Percival
11/14/13 Jussi Piitulainen
11/14/13 Dan Christensen
11/14/13 Peter Percival
11/14/13 Dan Christensen
11/14/13 Bart Goddard
11/16/13 gnasher729
11/14/13 Bart Goddard
11/14/13 Dan Christensen
11/14/13 Peter Percival
11/14/13 Dan Christensen
11/14/13 quasi
11/14/13 Dan Christensen
11/14/13 quasi
11/14/13 Dan Christensen
11/14/13 Brian Q. Hutchings
11/15/13 Dan Christensen
11/15/13 Bart Goddard
11/15/13 Dan Christensen
11/15/13 Bart Goddard
11/15/13 Dan Christensen
11/15/13 Bart Goddard
11/15/13 Dan Christensen
11/15/13 Bart Goddard
11/15/13 Dan Christensen
11/15/13 Bart Goddard
11/16/13 Bart Goddard
11/16/13 Dan Christensen
11/16/13 Bart Goddard
11/16/13 fom
11/15/13 Dan Christensen
11/14/13 Dan Christensen
11/14/13 Peter Percival
11/14/13 Peter Percival
11/14/13 Robin Chapman
11/14/13 Peter Percival
11/15/13 Robin Chapman
11/15/13 Peter Percival
11/17/13 Marshall
11/14/13 Bart Goddard
11/14/13 Dan Christensen
11/14/13 Bart Goddard
11/14/13 fom
11/13/13 David C. Ullrich
11/13/13 Dan Christensen
11/13/13 Peter Percival
11/13/13 Dan Christensen
11/13/13 Peter Percival
11/13/13 Dan Christensen
11/15/13 Shmuel (Seymour J.) Metz
11/15/13 Jussi Piitulainen
11/25/13 Shmuel (Seymour J.) Metz
11/26/13 Jussi Piitulainen
11/16/13 gnasher729
11/17/13 Dan Christensen
11/17/13 Peter Percival
11/17/13 Dan Christensen
11/17/13 Peter Percival
11/17/13 Dan Christensen
11/19/13 Dan Christensen
11/19/13 LudovicoVan
11/19/13 Dan Christensen
11/29/13 Dan Christensen
12/2/13 Brian Q. Hutchings