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Infinity means there is an infinite amount of something. Unsing same logic, infinity to the power of anything is infinity itself, because there is nothing larger than infinity. My friend proposed this thing to me. I know it should be wrong, but I can't really tell.
$$\infty^\infty=\infty$$ $$\infty^1=\infty$$ $$\implies\infty^\infty=\infty^1$$ $$\implies\infty=1$$

What is the catch here?

Thanks!
Max0815

Max0815
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  • Those operations are not defined for $\infty$; otherwise you could argue $\infty=\infty+1$ so $0=1$; cf. answers to this question – J. W. Tanner Aug 20 '19 at 00:47
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    There are multiple things that mean "an infinite amount." Some are numbers, some are not. Among the infinite numbers, some are larger than certain other numbers. If $\infty$ is a number and if $\infty^\infty$ means anything at all, I would expect that $\infty^\infty>\infty.$ – David K Aug 20 '19 at 00:53
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    We may extend some arithmetic operations to the extended real line $\overline{\mathbb{R}}=\mathbb{R}\cup{-\infty,+\infty}$ so as to correctly reflect their limiting behaviors, but such extensions invalidate some familiar algebraic rules that are true in finite regime. Your friend's fallacy is one such example, since $\infty^x=\infty$ for all $x>0$ means that you cannot invert exponential with infinite base. In order to retain such nice algebraic properties, we have to add a plethora of infinitely large numbers of different quality of infinitude, as in hyperreal numbers or surreal numbers. – Sangchul Lee Aug 20 '19 at 00:58
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    As mentioned in SmileyCraft's answer below, just because $1^2=1^3$, that doesn't mean that $2=3$. – JRN Aug 20 '19 at 01:01
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    Why vote to close this? The OP heard a strange idea, and instead of brazenly accepting it or refuting it, they sought more clarity from a reputable source. It should be encouraged. – The Count Aug 31 '19 at 00:12

2 Answers2

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That $a^b=a^c$ implies $b=c$ only holds for positive real numbers and $a\neq1$. This excludes for example complex numbers, but also infinity, which is not even a number. You can define arithmetic with infinity, but you can not expect this to obey the usual rules of algebra.

SmileyCraft
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Work on the extended real line. Then the following manipulations are okay: $$\begin{split} \infty^{\infty}&= \infty^1 \\ \infty\ln \infty &= 1 \ln \infty \\ \infty (\infty) &= 1(\infty)\text{.} \end{split}$$ But we cannot divide $(\infty)$ from both sides of the last line to conclude $\infty\stackrel{!}{=}1$.

Generally speaking, we can't expect expressions like $\infty-\infty$ or $\infty/\infty$, or other indeterminate forms to have a meaning in the arithmetic of the extended reals.

K B Dave
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