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
On
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.
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.