>When talking about empty products, it's particularly nice to
>consider Taylor's series. The kth term involves
>
>f(k)(a), k!, and (x - a)k.
>
>When k happens to be 0, _all three_ of these quantities are
>empty "products".

You are going to answer a question I did not ask. All I said was that I do not know of any Calculus book that took up the matter of 00 = 1.

>>>Of course, it's no mistake to have Calculus in mind. The
>>>mistake is to think that Calculus would keep us from
>>>defining 00 as it should be!

Who does?

>>Again, I see your point and enjoy your argument but I can't
>>accept the value judgement concealed behind "... as it
>>should be."
>
>If it would make you any happier,

This is besides the point. I am quite happy as it is. And my happiness or its degree is irrelevant to the discussion.

>Consider the statement "23 = 8 is as it should
>be." Do you really think that there is some unacceptable
>value judgment concealed therein? I don't.

I do not.

>Similarly,

???

>I don't hesitate to say that 00 = 1 is
>as it should be.

I've been reading with my 6 year old a small book "Why did not I say that?" on a matter of a suitable repartee under various circumstances.

The thing that just leaps out of my mouth is "Do you ever?" - Sorry for that. But seriously, if the above is not a value judgement, I do not know what is.

>To me, having 00 undefined is just such a thorn.
>Sure, we could live with it and do well enough. But why not
>remove the thorn?

By all means. Please go ahead.

>You raise pedagogical concerns. Clearly, such concerns are
>valid. But of course we know that what is _pedagogically_
>nice cannot dictate what is _mathematically_ correct.

We are not going to start a discussion on mathematical correctness, are we?

>What
>is mathematically correct is determined first; only
>subsequently may we consider how that can best be taught.
>
>But in fact, I consider having x0 = 1 for _all_ x
>to be not only a mathematical necessity, but also
>pedagogically easier to handle than leaving 00
>undefined. After all, if we explain well why _any_
>0th power must be 1

This you did explain really very well.

>, then we are done.

But if only life were that easy

>Period.

Ellipses.

>undefined. But if you know of any such reason, Alex (or
>anyone else), please give it!

There is none. Now what?

As I said, I liked your argument and the notion of an empty product. All you need now is a magic wand.

I wish you well,
Alex