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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 ]
 Virgil Posts: 8,833 Registered: 1/6/11
Re: The ambiguity of 0^0 on N
Posted: Sep 20, 2013 3:42 PM

Dan Christensen <Dan_Christensen@sympatico.ca> wrote:

> On Friday, September 20, 2013 8:54:45 AM UTC-4, Peter Percival wrote:
> > Dan Christensen wrote:
> >

> > > fom, how about some honest toil to prove 0^0=1? Choose any axioms or
> > > definitions you like. Just not 0^0=1 itself or anything like x^0=1 for
> > > all x in N. That would be just too easy! ;^)

> >
> >
> >
> >
> >
> > I think the time has come for you to say what axioms and definition
> >
> > _you_ are using. Let's take first order logic with = for granted, what
> >
> > primitive extra-logical symbols do you have and what extra-logical
> >
> > axioms govern them? If ^ isn't among the primitive extra-logical
> >
> > symbols how is it defined in terms of them?
> >

>
> See my proof. The link again: http://dcproof.com/Ambiguity-of-0-to-the-0.htm
>
> In lines 1-14, I list the definitions and properties of the natural numbers
> required for the proof. They are treated as axioms for the purposes of this
> proof, but you can think of them as theorems or definitions.
>
> In lines 15-17, I give the axioms/definitions for ^.
>
> In lines 18-41, I prove that x^1=x for x=/=0.
>
> In lines 42-83, I prove that x^0=1 for x=/=0.
>
> In the remainder of the proof, I attempt to fill in the gaps -- assigning
> values to 0^0 and 0^1. I show that if we assume the Power of Products Rule
> (pretty standard stuff), then 0^1 must be 0 and 0^0 must be either 0 or 1.
>
>

> >
> >
> > You have been shown proofs that 0^0 = 1 and you reject them,

>
> See my reply to fom.
>

> > so what
> >
> > theory (= logical system) of natural numbers do you wish your
> >
> > interlocutors to work in? Spell it out in detail.
> >

>
> I haven't developed the entire system of natural numbers here. I have just
> listed some number-theoretic results (lines 1-14) that are required for my
> proof. They are stated in a formalized version of the language of everyday
> mathematics. If you take exception to any of them, let's discuss it.
>
> But you need not use my system, although that would be nice. Use whatever
> system you like. Improvise one on the spot if you must. Just don't assume
> what you are trying to prove as you have repeatedly tried to do here.
>

> >
> >
> > That a theory of numbers can be cooked up in which 0^0 = 1 is a wff that
> >
> > can't be proved is clear. But so what?
> >

>
> There is nothing weird about my number system. Again, the notion that 0^0 is
> undefined is not some new, radical idea. It has been around for centuries and
> is really quite mainstream. (See that Wiki article I cite above.) If you
> polled all the world's math instructors, I am sure that the vast majority
> would agree that 0^0 should be left undefined.
>
> Again, if you disagree with any of my assumptions or definitions (lines
> 1-17), let's discuss it.
>
> Dan

I can see that
f(x,y) = x^y is continuous on domain {(x,y): x > 0 and y > 0}
and can even be extended continuously to both
{(0,y): y > 0} and
{(x,0): x > 0}
but there is no way to extend that function CONTINUOUSLY to
{(x,y)| x > 0 and y > 0} to {(x,y)| x >= 0 and y >= 0}.

This suggest that for real functions like
f(x,y) = x^y on {(x,y): x > 0 and y > 0}
there is no universally acceptable limit value at f(0,0).
--

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