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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.

Sam Frank
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    In some sense, mathematics can be reduced to a finite list of symbols, and a list of rules for ways you are allowed to combine the symbols. $3+3$ makes sense in our system, but $3+3+$ does not make sense. Same with $\infty - \infty$. – D_S Feb 27 '21 at 02:18
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    mostly they mean that certain limits, such as $f(x)^{g(x)}, $ may have different values depending on choices made in $f,g$ – Will Jagy Feb 27 '21 at 02:18
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    I doubt that there exist interesting insights on this - the idea that an indeterminate form indicates some fundamental flaw seems silly. – David C. Ullrich Feb 27 '21 at 02:18
  • I find no discomfort in the existence of indeterminate forms and do not see them as "flaws" and see no justification in thinking our current system is flawed. Everything follows from what definitions and axioms we chose. – JMoravitz Feb 27 '21 at 02:27
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    I see the argument that mathematics is flawed because of indeterminate forms the same as saying the post office is flawed because if I try to mail a letter in a blank envelope with no address written on it they don't know who to send it to. "The postmen should have been able to figure it out" How? In the same way, being told $f(x)\to 1$ and $g(x)\to \infty$ and asking what the limit of $f(x)^{g(x)}$ is... it is like an envelope with no address... The fault lies in the question, not in the system. – JMoravitz Feb 27 '21 at 02:34
  • Is "object" a noun or a verb? Is the fact that an isolated word can be ambiguous a flaw in the language? We need a context to resolve that question. An indeterminate mathematical expression such as $ \ 0/0 \ $ or $ \ 0^0 \ $ indicates that numbers are being used with an operation in a way that cannot be resolved without including something about how the expression arose, so that a meaning might be assigned. (And using $ \ \infty \ $ in an expression just makes matters worse, since it is not a number.) –  Feb 27 '21 at 02:37

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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?

Graviton
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