What does $\aleph_{\aleph_{\aleph_0}}$ stand for?

Question

My guess is that it's the cardinality of $\omega_{\omega_{\omega}}$. Since $\aleph_0$ is the cardinality of $\omega$ and $\omega$ is a limit cardinal, so it must be that $\aleph_0=\omega$. (Is this true in general?) Then I can just substitute the subscript $\aleph_0$ with $\omega$, and I do the same thing again. Is this correct?

Answer 1

Without any further context, the cardinality of $\omega_{\omega_\omega}$ seems to be the most reasonable guess at the meaning, but the notation is not standard. The subscript in $\aleph_{n}$ is supposed to be an ordinal number, not a cardinal number, so the standard notation for this would be $\aleph_{\omega_\omega}$.

If this interpretation makes sense in context, I would assume it is just a misnaming of $\aleph_{\omega_\omega}$, unless you're positively sure the $\aleph_{\aleph_{\aleph_0}}$ notation comes from someone who definitely knows what they're talking about.

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(If you're asking due to the comment on your earlier question, I'm pretty sure Duncan just meant $\omega_{\omega_\omega}$ -- but not because there's anything special about that ordinal; any other large-looking ordinal could have been used to make the point he was making).

Answer 2

It would be more appropriate to write this as "$\aleph_{\omega_\omega}$" - the index of an expression is supposed to be an ordinal, not a cardinal. And this is indeed the cardinality of the ordinal $\omega_{\omega_\omega}$.

However, cardinals are usually defined as initial ordinals, so e.g. $\aleph_0=\omega$; it's just that we often use two different names for these entities, to distinguish between their roles in a mathematical argument. So technically the expression is fine as written, if a bit odd.