Are indeterminate forms a sign that our mathematics system is flawed?

Question

This is much more philosophical and opinion-based than many of the other questions asked here, but yeah, pretty much what the title says. Sometimes I think that we have this system of mathematics simply because we could count with whole numbers and it sort of snowballed, but then indeterminate forms like $1^\infty$ or $0^0$ being ambiguous make me think that maybe our system has fundamental flaws. I don't know. I'm no mathematician, just an undergrad, so if anyone has some interesting insights on this, I'd be happy to hear them.

Answer 1

Obligatory "not a full answer but too long for a comment". That being said, this is a soft-question so no answer may be 100% satisfying.

Is mathematics flawed? No.

Is mathematics incomplete? Absolutely.

Gödel's incompleteness theorems and the Halting problem are infamous proofs that systems equipped to handle finite logic/arithmetic (where mathematics has "snowballed" from), will never be able to handle the paradoxical nature of infinity & recursion consistently & completely.

It seems to me that indeterminate forms are indeterminate not because mathematics is flawed, but because an infinity arises. Mathematics can often only at best treat the supertask that is infinity, as a limit; which may inherently underestimate the complexity of the true nature of infinity.

A good example are infamous divergent sums, such as $1+2+3+4+\dots$, where our finite math clearly indicates a divergent sum. Yet, looming in shadows is the strange pseudo-convergence to $-\frac{1}{12}$, which could only ever be correct if our understanding of infinity were incomplete.

On top of such, we often don't even know which infinity we're referring to when using it in a sentence or an expression. Do we want to treat $\infty$ in $1^\infty$ being raised to a limit or having a number-like value? Ordinal infinity or cardinal infinity? Countable infinity or uncountable infinity? Do these questions even make sense in this context?