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The whole point of distinguishing some forms as "indeterminate" is to work around the fact that you're trying to conduct operations on the real numbers that are

The whole point of distinguishing some forms as "indeterminate" is to work around the fact that you're trying to conduct operations on the real numbers that are not defined under the field axioms of the real numbers (or the axioms of the extended reals [-inf,inf]). That's where its essence is rooted; that's why this whole affair exists -- the fact that 0/0, 0^0, 0xinf, inf-inf, etc. are not well defined by our axioms. This actually relates to all three examples I've presented where the 0^0=1 convention fails. It should be treated as an indeterminate form in limits because it's not well-defined by the axioms of the real numbers; it's also not well-defined by the axioms of the hyperreals, but division of infinitesimals is well-defined in the hyperreals, which gives us an alternate method of computing limits that avoids the "indeterminate form" entirely. Let me reiterate: 0^0 is not defined under the field axioms of the real numbers. The choice to define it as 1 is a convention which makes certain math easier, in certain areas of mathematics. It is by no means a universal convention; it is by no means the one and only correct definition of 0^0. You continue to argue for the convention, but miss the larger point that it is a convention which is chosen for convenience, and which is not always appropriate.

xyzzyz 15y ago
Exponential function is not defined by the axioms of real numbers, as opposed to addition and multiplication, so this point is irrelevant -- you can define it in any way you want. There's no inconvenience in defining 0^0 = 1, apart from misunderstanding the concept of limits by some people. Defining 0^0 = 1 is universal convention -- people either do it like this, or do not define 0^0 at all, which I'm fighting against. The whole point of distinguishing some forms as "indeterminate" is to work around the fact that you're trying to conduct operations on the real numbers that are not defined I am not. Are you? Let me reiterate: the whole concept of "indeterminate forms" (which, I repeat, is not formal at all) stems from misunderstanding the process of taking limits.
- lotharbot 15y ago
  > "people either do it like this, or do not define 0^0 at all, which I'm fighting against." I think it's silly to fight against it. There are circumstances in which leaving it undefined is good, and in which trying to define it as 1 would lead to either misunderstandings (in the case of beginners doing limits, a case you are too quick to dismiss) or actually incorrect (an equivalent problem in the hyperreals could violate the transfer principle). It's a broad convention, but it is not universal, and it shouldn't be.

xyzzyz 15y ago

Exponential function is not defined by the axioms of real numbers, as opposed to addition and multiplication, so this point is irrelevant -- you can define it in any way you want. There's no inconvenience in defining 0^0 = 1, apart from misunderstanding the concept of limits by some people. Defining 0^0 = 1 is universal convention -- people either do it like this, or do not define 0^0 at all, which I'm fighting against. The whole point of distinguishing some forms as "indeterminate" is to work around the fact that you're trying to conduct operations on the real numbers that are not defined I am not. Are you? Let me reiterate: the whole concept of "indeterminate forms" (which, I repeat, is not formal at all) stems from misunderstanding the process of taking limits.

lotharbot 15y ago
> "people either do it like this, or do not define 0^0 at all, which I'm fighting against." I think it's silly to fight against it. There are circumstances in which leaving it undefined is good, and in which trying to define it as 1 would lead to either misunderstandings (in the case of beginners doing limits, a case you are too quick to dismiss) or actually incorrect (an equivalent problem in the hyperreals could violate the transfer principle). It's a broad convention, but it is not universal, and it shouldn't be.