Strange Cardinal Notation

Question

Is anybody aware of any references which may justify this notation? $$ \LARGE \aleph^{\aleph^{\aleph\dots}_{\aleph\dots}}_{\aleph^{\aleph\dots}_{\aleph\dots}} $$ In particular, using Cardinals for indexing) mathematically (with ZFC)? What about it if we change all subscripts to be Ordinals?

Answer 1

In some relatively introductory books (I think Jech–Hrbacek is one) you will find $\aleph_{\aleph_0}$ as a notation, and I'd guess also $\aleph_{\aleph_{\aleph_0}}$ and so on.

I find this to be a problematic notation. We use $\aleph_0$ to denote the cardinal, and that sets up the context for arithmetic as cardinal arithmetic. In particular, what would be the successor of $\aleph_{\aleph_0}$? It would be $\aleph_{\aleph_0+1}$. But we know that $\aleph_0+1=\aleph_0$. Does that mean the cardinal is its own successor?

Yes. That is a problem. And it is easily solved by sticking to the actual indexing by ordinals. Namely, $\aleph_\omega$ and $\aleph_{\omega_\omega}$ and $\aleph_{\omega_{\omega_\omega}}$, and so on.

This leads us to interpret $\aleph_{\aleph_{\aleph_{\ddots}}}$ as $\aleph_{\omega_{\ddots}}$, or more accurately as the least $\alpha$ such that $\alpha=\omega_\alpha$. This is known as a fixed point. And $\sf ZFC$ proves that these fixed points exist, and in some sense make up "most of the cardinals" (indeed, most of the ordinals, since "most ordinals are cardinals" in the same sense).

So there is a first $\alpha$ such that $\alpha=\omega_\alpha$, and a second, and there is $\alpha$ which is the $\alpha$th one with that property too. And so on.

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Moving on to the exponent, $\aleph_\alpha^{\aleph_\beta}$ makes a lot of sense. It's just cardinal exponentiation. If $\alpha=\beta$, namely $\aleph_\alpha^{\aleph_\alpha}$, we just get $2^{\aleph_\alpha}$. What is the exact $\beta$ such that $\aleph_\beta=2^{\aleph_\alpha}$? That we cannot determine from $\sf ZFC$ alone. But the notation itself makes sense.

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Moving on to the image itself. How would I interpret that? Well, I wouldn't. I would probably ask whoever suggested that cardinal to come up with a rigorous definition. For example, the first fixed point $\alpha$ is the limit of the sequence $\alpha_0=\omega$ and $\alpha_{n+1}=\omega_{\alpha_n}$.

So for that notation to make sense, it should probably be written as a limit of such sequence. But yeah, the idea of such cardinals existing can boggle ones mind.