
Re: The ambiguity of 0^0 on N
Posted:
Sep 18, 2013 12:10 PM


Dan Christensen wrote: > On Wednesday, September 18, 2013 11:08:01 AM UTC4, Peter Percival wrote: >> Dan Christensen wrote: >> >> >> >>> >> >>> More formally, exponentiation can be defined as a binary function on >> >>> the set of natural number N such that: >> >> >> >> Can be but isn't, except by you. Everyone else, when defining a >> >> function on N, will start where N starts, viz at 0. >> >> >> >>> >> >>> (1) Ax in N: x^2=x*x >> >>> >> >>> (2) Ax,y in N: x^(y+1) = x^y * x >> >>> >> >>> It can then be shown that: >> >>> >> >>> (1) Ax in N:(x=/=0 => x^1=x) >> >>> >> >>> (2) Ax in N:(x=/=0 => x^0=1) >> >>> >> >>> (3) Ax,y,z in N:(x^(y+z) = x^y * x^z) => 0^1=0 /\ (0^0=0 \/ 0^0=1) >> >>> >> >>> (Formal proof to follow.) >> >>> >> >>> 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 >> >> >> >> 0^0 = 1 _can_ be formally proven, if you use the right definition. >> > > I think you mean it can be "proven" if you define it as such.
So a proof that follows pretty quickly from the definitions is not a proper proof, but a longer proof is kosher? Suppose this is offered as a proof of a formula D:
1. Some definition or other. 2. Formula A justified by it following from 1. 3. Formula B justified by it following from 2. 4. Formula C justified by it following from 3. 5. Formula D justified by it following from 4.
Would you say "that is a proof that is five lines long, that's enough for validity, I accept it as a proof of D"? Suppose this is offered as a proof of a formula A:
1. Some definition or other. 2. Formula A that follows from 1.
where those two lines are precisely the same as the first two lines in the first proof. Would you say "that is not a valid proof of A because it is only two lines long"? If the second proof isn't valid, then neither is the first, so I hope you would not say that. But your dislike of
1. x^0 = 1 (clause in a definition) 2. 0^0 = 1 from 1. by the substitution x := 0
seems to be because 0^0 = 1 is an immediate consequence of the definition. In other words you think short proofs aren't proofs, but long ones are.
> >> >> >> If your definition of ^ doesn't tell you what 0^0 is (remember, we're >> >> talking about the naturals here) then the definition is incomplete. >> > > Perhaps in the sense that it doesn't define a unique function  just a function on N with certain properties. > >> >> >> If you want to define ^ on the positive integers only and then claim >> >> that 0^0 isn't thereby defined, then you'd be right. But it is _you_ >> >> who claim to be talking about N. >> > > It should clear from the context that I am talking about the natural numbers including 0.
Yes and no. You deny the first line of
x^0 = 1 x^{y+1} = x*x^y
so you seem to exclude 0.
People who know far more than you or I have defined things in particular ways for good reasons. Perhaps you don't believe that, perhaps you think you know best. Do you? If so you're deluded.
 Sorrow in all lands, and grievous omens. Great anger in the dragon of the hills, And silent now the earth's green oracles That will not speak again of innocence. David Sutton  Geomancies

