1^∞, 0^0 and others on the real projective line

[Diffy]

Hurkyl is exactly right.

I said earlier that these things are not defined, because when you define them you get contractions.

Wouldn't you agree that 0≠0 is a contradiction?

This is why we leave these things undefined.

 

[Hurkyl]

Hurkyl is exactly right.

I said earlier that these things are not defined, because when you define them you get contractions.

The reason they aren't defined is because it's not useful to define them.

It's not useful to define them, because if you did, it wouldn't have many useful properties.

You only get a contradiction if you both define them and assume that the definition has useful properties.

 

[MTannock]

The reason they aren't defined is because it's not useful to define them.

It's not useful to define them, because if you did, it wouldn't have many useful properties.

You only get a contradiction if you both define them and assume that the definition has useful properties.

While I agree in principle, I've sometimes found it useful to define 0^0 as 1.

 

[Mark44]

While I agree in principle, I've sometimes found it useful to define 0^0 as 1.

Hurkyl was referring to ∞/∞ and 0/0.

A lot of calculators agree with you on 0^0, and produce 1 as a result. This doesn't have anything to do with the real projective line or the extended real number line, though.

 

[Hurkyl]

Actually, 0^0 is relevant here. It's also a wonderful illustration of some of the relevant problems.

On the one hand, is not useful to give a value to 0^[/color]0. On the other hand, 0^[/color]0 is not only equal to 1, but it's not even a special case.

What changed from one side to the other is that ^[/color] refers to the continuous exponentiation operator or its continuous extensions to things like the extended and projective real lines. ^[/color], however, is being used to some sort of algebraic exponentiation operator; examples have domains including things like the base from any ring and exponent from the natural numbers. The exponentiation operation appearing in a power series is ^[/color].