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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 12:39 AM

On Wednesday, September 18, 2013 7:20:33 PM UTC-4, federat...@netzero.com wrote:
> On Wednesday, September 18, 2013 9:53:13 AM UTC-5, Dan Christensen wrote:
>

> > Thus, if the Product of Powers Rule is to hold on N, 0^0 will be ambiguous -- being either 0 or 1.
>
>
>
> To address the problem of how to define 0^0, one should first set the playing field. For example, one might frame the question this way: what [continuous/differentiable/analytic/C^infinity/or/etc.] function, if any, satisfies f(x, y) = x^y when x and y are not zero, and has (x, y) = (0, 0) in its domain.
>

Using only natural number arithmetic and assuming PPR, you can prove that 0^0=/=2. Until you can prove either 0^0=/=0 or 0^0=/=1 in this way, 0^0 remains ambiguous.

>
>
> If the criterion is "continuity", then the list of candidates for f(0, 0) consists of all the limits of f(x, y) as (x, y) -> (0, 0) over (say) all paths.
>
>
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> Then, every number is in the candidate list. That is: every COMPLEX number.
>
>
>
> If the problem is framed like this: what choice for 0^0 yields, say, a realization of the Curry-Howard correspondence, then the answer is unambiguously 1. The correspondence is best seen by laying out the tautologies of the logic of the "if-then" operator alongside the rules for exponentials.
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>
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> Then, one has the following correspondences -- valid for both classical propositional logic and intuitionistic logic:
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>
>
> (Let X -> Y denote the "if X then Y" conditional statement.)
>
> A^{B+C} = A^B A^C <==> (B or C) -> A iff (B -> A) and (C -> A)
>
> A^0 = 1 <==> (false -> A) iff true
>
> A^{CB} = (A^B)^C <==> (C and B -> A) iff C -> (B -> A)
>
> A^1 = A <==> (true -> A) iff A.
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> (AB)^C = A^C B^C <==> (C -> A and B) iff (C -> A) and (C -> B)
>
> 1^C = C <==> (C -> true) iff C
>
>
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> The equation 0^0 = 1 then corresponds to the tautology (false -> false) iff true.

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