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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 ]
 Rotwang Posts: 1,685 From: Swansea Registered: 7/26/06
Re: The ambiguity of 0^0 on N
Posted: Sep 19, 2013 9:31 AM

On 19/09/2013 06:23, Dan Christensen wrote:
> On Wednesday, September 18, 2013 7:11:53 PM UTC-4, Rotwang wrote:
>> On 18/09/2013 15:53, 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. Unless one of these alternatives can be formally proven
>>
>> Obviously if you start of with a "definition" of ^ which leaves 0^0
>> undefined,

>
> Actually, I start with a definition which leaves out any explicit mention at all of exponents of 0 or 1. I define only exponents greater than 1. Starting with this definition, I prove that x^0 = 1 and x^1=x for x=/=0.

But your definition leaves 0^0 undefined. So nothing can be proved about

> I also prove that if you want to extend the Product of Powers Rule to all of N, then we must have 0^1=0 and either 0^0=0 or 0^0=1.

Yes, I know. "If you want to extend the Product of Powers Rule to all of
N" is an additional assumption. But what I'd like to know is, what's so
implies that x^0 = 1 for x != 0 - let's call this the Power of Zero
Rule. Why do you want to extend the Product of Powers Rule to all of N,
but not the Power of Zero rule?

> Until we are able to prove or disprove one of these alternatives, we should probably leave 0^0 undefined.

Why? Note that I'm not asking you to repeat your argument that if one
defines ^ a certain way one finds that there are two different functions
why anyone should define ^ the way you do in your OP, rather than the
much simpler and more common way that it's usually defined, which has
the additional benefit that there is a unique function that satisfies
it, and that PPR can be derived from it, rather than just assumed.

> It seems you can only prove something about 0^0 if PPR is extended over all of N.

Or if you use a sensible definition of ^.

>>> the prudent course
>>
>> You keep saying this. Why?

>
> Because I believe it is true.

Why?

>> What negative consequences do you imagine
>> 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.

But the consequences from a calculation that results in "undefined" when
it should be 1 could be similarly catastrophic. Why do you imagine that
the consequences of 0^0 returning 1 when it "should" be 0 would be any
more catastrophic, or more likely, than the consequences of 0^0 not
returning 1 when it "should" return 1?

>> Why do
>> you imagine these consequences are more serious, or more likely, than
>> the obvious negative consequences of changing the behaviour of C, Java,
>> Javascript, Python, Perl, Ruby, Lisp, Haskell and probably most other
>> programming languages, or of changing a mathematical definition in such
>> a way as to render many well-known theorems false?

>
> The most well known theorem that makes use of 0^0 is probably the Binomial Theorem. And it can easily be restated to avoid its use, e.g. by stating at the outset that (x+0)^n = x^n, etc.

Why yes, rewriting every textbook that states the binomial theorem, any
theorem about continuity of polynomials, power series, cardinal
exponentiation or any number of other mathematical results, as well as
changing the behaviour of all those programming languages I mentioned,
would almost certainly be easier than getting you to acknowledge that
weight than those of the people who write mathematics textbooks and
design programming languages. But that doesn't answer my question. Why
do you imagine the negative consequences of continuing to define 0^0 = 1
are more serious, or more likely, than those of changing the definition
on the whims of Dan Christensen? In pondering this question, it's worth
bearing in mind that time we had every textbook and web page about
category theory rewritten after you sort-of read the introduction to a
wiki page on the subject and decided that the conventional definition of
morphism was wrong. And then we had them all changed back again a couple
of weeks later, when you learnt a bit more about the subject and decided
that the definition was right after all. Remember that, Dan? I mean, as
far as we can tell it only cost the world a few thousand man-hours and
several hundred acres of rainforest that had to be cut down to provide
the paper for all those new editions, but the full repercussions may not
be known for some time.

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