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?
$\infty$ 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 $1^\infty$.