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We see the common questions in math going through highschool. 1^infinity, undefined, why? Because infinity isn't defined in the reals? So we reorganize the question to include a limit, but we still use the symbol infinity in the limit definition. When working in the reals, what are we calling that infinity? Is it somehow different than the infinity we use in the extended reals? How so, exactly?

So I guess my question is, what is infinity in that context? I feel like if you have a symbol, and claim it's an invalid symbol to use when working in the reals, why do we continue to use the symbol when working on a problem defined in the reals?

Why can't we call infinity a number in the reals, but still place restrictions on what operations are defined when it's being used.

I mean, 0 is uncontroversially considered a number, right? But 0/0 is undefined. 1/0 is undefined.

Why isn't infinity considered a number in the reals? That way, 1^infinity is defined. It's 1. 2^infinity? it's infinity. Infinity + 5? Infinity. Infinity / 0? Who knows.

It just feels bizarre to say infinity isn't a number, when talking we implicitly know the exact meaning somebody is getting at.

To me it's kind of like saying -1 isn't a number because it's not a natural number. Or i isn't a number because it's not defined in the reals. What's with the bias with the formal definition of reals, and why can't we drop the tired talking point that infinity isn't a number.

Since there isn't a context where 1^infinity could possibly resolve to any other value than 1, why can't we call it 1 and move on with our lives?

Mauro ALLEGRANZA
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SomeGuy5432
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1 Answers1

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$\infty$ is a number in the extended reals $\overline{\mathbb{R}}$. However, $\infty$ does not behave like any quantity in $\mathbb{R}$ or $\mathbb{C}$ - in particular, we cannot extend the usual operations on $\mathbb{R}$ or $\mathbb{C}$ to $\overline{\mathbb{R}}$, since quantities such as $\infty-\infty$, $0\times\infty$, or $\infty/\infty$ cannot suitably be defined as to be consistent with how we would expect $\infty$ to behave.

The symbol $\infty$, when used in the context of limits, has a very precise definition independent of any actual algebraic manipulations: via $\varepsilon-\delta$ or $\varepsilon-N$ definitions. For example, we say that $\lim\limits_{n\to\infty}a_n=L$ if for all $\varepsilon>0$ there exists $N\in\mathbb{N}$ such that $\forall n\geq N$, $|a_n-L|<\varepsilon$; we are not using any actual properties of $\infty$ in this definition, and $\infty$ is merely a symbol.

Also, $1^\infty$ can resolve to something other than $1$ in a limiting case - this is exactly why we cannot define such quantities, even in $\overline{\mathbb{R}}$. $(1+x)^{1/x}$ does not approach $1$ as $x\to0$, for example, despite being of the form $1^\infty$.

csch2
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  • When you say " we cannot extend the usual operations on ℝ or ℂ to ℝ⎯⎯⎯⎯⎯, since quantities such as ∞−∞, 0×∞, or ∞/∞ cannot suitably be defined as to be consistent with how we would expect ∞ to behave." why can't that argument be used against 0? 0 has undefined operations, so why can't infinity? I would also argue that the limit representation of what you had before means something completely different than 1^infinity. You're taking the limit of two different spots of a function that each converge to a different value. And the operations of those values isn't the same as the limit of the func – SomeGuy5432 Apr 30 '20 at 23:58
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    An operation such as $-$ needs to be a function, i.e. it would need to be a map $\overline{\mathbb{R}}\to\overline{\mathbb{R}}$. But any definition we gave for $\infty-\infty$ would necessarily be problematic - of course we want $1+\infty=\infty$, but then $1+\infty-\infty=1$ and $\infty-\infty=0$ would imply $1=0$ with the usual rules for addition and subtraction on $\mathbb{R}$. The rules we have work fine for $0$, as you can tell from everyday experience with $\mathbb{R}$, with the notable exceptions of the indeterminate forms. – csch2 May 01 '20 at 00:02
  • To me, that's like saying 0 isn't a number because we have 5*0=0, so we would want 0/0 = 5. infinity-infinity would be undefined. 1 + infinity -infinity would still be undefined. Since reals isn't defined as a system of numbers where all operations between any numbers are defined, I don't understand the exclusion of infinity. – SomeGuy5432 May 01 '20 at 00:05
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    It is perfectly fine to say $\infty$ is a number, so long as you don't expect it to behave like other numbers such as $2$ or $\pi$. Simply put, you can't extend the rules of arithmetic on $\mathbb{R}$ to include $\infty$ in a way that doesn't introduce contradictions, just like we don't try to define $0/0$ for the exact reason that it introduces a contradiction. If you would like, it is more convention not to include $\infty$ in $\mathbb{R}$, for the reason that $\mathbb{R}$ satisfies certain nice properties without $\infty$. – csch2 May 01 '20 at 00:09
  • For example, by excluding $\infty$, you preserve the Archimedean property: for each $x\in\mathbb{R}$, there exists $n\in\mathbb{N}$ such that $n>x$. This is an important property lost if $\infty$ is added to $\mathbb{R}$. – csch2 May 01 '20 at 00:10
  • How does defining infinity + 1 = infinity introduce a contradiction? infinity + N, where N is a natural number, would act in a similar way as 0*N. We're comfortable identifying when 0 introduces an undefined result, why can't we carry that same thinking over to infinity? What exactly does it break? I don't think your example answers that. – SomeGuy5432 May 01 '20 at 00:11
  • Ah okay. So I can definitely see properties of the reals changing as a consequence of introducing infinity as a number of the reals. But then my question turns into, so what? Yeah every other number in the reals would be less than infinity, but how does that really change anything, other than having one extra caveat? Is there some greater definition of reals that I don't understand? Or is this mostly a point of history, where infinity as a concept was formally defined after the reals were already defined? Feels like infinity is more or less excluded out of ignorance more than anything. – SomeGuy5432 May 01 '20 at 00:19
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    Another very important property of $\mathbb{R}$ is being a field, an important type of algebraic structure. If you include $\infty$, then you lose a very large amount of algebraic content - you go from having the rich structure of a field to not even having additive inverses for each element. – csch2 May 01 '20 at 00:25