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Note: I read something about this on the internet somewhere once, my logic could be 100% flawed.

Is $(\aleph_0)^{\infty} = \infty^{\infty}$?

Suppose the number $\infty^{\infty}$. Assuming $\infty$ in this sense has a cardinality of $\aleph_0$, could it be represented as $(\aleph_0)^{\infty}$?

As an extension, because $\aleph_0$ has a cardinality in it's set (some sort of countable infinities), is $(\aleph_0)^n = \aleph_0$?

It seems to be that $\infty^{\infty} > \infty$, and by extension, $\infty^{\infty} > \aleph_0$, but at the same time, if $\aleph_0$ is the $1$ of it's own set, $(\aleph_0)^\infty = \aleph_0$ the same way $1^1 = 1$.

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    $\infty$ and $\aleph_0$ don't go together. They occur in separate contexts, but you can't mix them like that. – 2017-01-21
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    Presumably, you mean cardinality. Carnality is covered in the rest of the internet. – 2017-01-21
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    Mathematical prurience? Go fourth and multiply? – 2017-01-21
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    @copper.hat, typo. – 2017-01-21
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    It would really be nice to know what you think $\infty$ means. In any event, the title asks about $\aleph_0$ and the body asks about $\aleph_0^{\infty}$. Which question do you mean? – 2017-01-21
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    @ThomasAndrews, I think the problem arises from my source of $\aleph_0$, I thought there were at least sort of the same thing – 2017-01-21
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    No, we don't use $\infty$ in general to refer to any cardinality. It is a symbol that just is not used in that context. It is used in calculus when writing $\lim_{x\to\infty}$ and $\int_{0}^{\infty}$ and other related cases - essentially, adding a point or points at infinity in a geometric way. But that $\infty$ never means a cardinality. – 2017-01-21
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    @ThomasAndrews I simply take $\infty$ to be larger than any real number. Thus, it isn't well defined in terms of cardinals. – 2017-01-21
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    Then what do you mean by $\infty^{\infty}$? Or $\aleph_0^{\infty}$? Those just are meaningless for this made-up meaning of $\infty$. – 2017-01-21
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    @ThomasAndrews I'm not the OP btw. – 2017-01-21

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$\aleph_0$ is the cardinal number of the set of naturals. If you are working in set theory it is well defined. $\infty$ is an informal concept usually used when you are using the reals, the integers, or something like that. It is not a number in any of those systems but used as something to take limits to, for example.

In set theory $\aleph_0^{\aleph_0}$ makes sense. It is the cardinality of the set of functions $\Bbb N \to \Bbb N$, for example. It is equal to $\mathfrak c$, the cardinality of $\Bbb R$. As $\infty$ is informal, I don't know what to make of $\infty^{\infty}$

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    +1, but I think it is misleading to call $\infty$ "informal". It is perfectly formal, in that it is a symbol we use with very precise meaning in certain contexts. And there are plenty of contexts where we treat it as a genuine mathematical object, an element of the set $[-\infty,\infty]$. It's just that we don't normally choose to define arithmetic operations involving $\infty$ (and in any case, in contexts where we do, it still doesn't make sense to compare it to cardinal numbers since it is not a cardinal number). – 2017-01-21
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    $\infty^\infty =\lim_{x\to\infty} x^x=\infty$ – 2017-01-21
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I have read the following chain of cardinal numbers in a Topology book

$1<2<3<......<\aleph_0<2^{\aleph_0}=c< 2^c < 2^{2^c}<.....$

Clearly, this chain has only one kind of countable infinity i.e. $\aleph_0$ and beyond this there is an infinite hierarchy of uncountable infinities which are all distinct from onf another.