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I stumbled over this: Infinite Set is Disjoint Union of Two Infinite Sets

but from my current knowledge, this is far over my hat.

At university, we have learned that $\aleph_0$ and $\aleph_1$ are different by Cantor's proof, and that there are more cardinalities based on this, especially the power sets of a set with cardinality $\aleph_i$ has cardinality $\aleph_{i+1}$ but the link above opens a tar pit of confusion with other possible cardinalities.

Where should I start reading?

Please note the tag [soft-question], I know this is not a "good" questions based on the standards here, but as of now, I can't do any better.

Gyro Gearloose
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    First, $\aleph_{i+1}$ usually shouldn’t be thought of as the cardinality of the power set of $\aleph_{i}$. This is essentially the generalized continuum hypothesis, which is independent of set theory. Second, if you want to learn more set theory, Halmos’s “Naive Set Theory” is an excellent place to start. Flip through the table of contents and see if it seems like a good fit. – Joe Mar 16 '25 at 19:06
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    To add to what @Joe said, you've likely seen the proof showing the difference between $\beth_1$ and $\aleph_0=\beth_0$, but not that between $\aleph_1$ and $\aleph_0$. – Y. Tamer Mar 16 '25 at 19:11
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    Also, for what it’s worth, the post you linked is difficult largely because it’s focused on set theory without the axiom of choice. When you abandon choice, the idea of “cardinality” becomes much more subtle. For example, it ceases to be a linear order. – Joe Mar 16 '25 at 19:11
  • @Y.Tamer The proof was between the cardinalities of $\bf N$ and $\bf R$. Cannot remember if any proof beyond that was given. – Gyro Gearloose Mar 16 '25 at 19:23
  • For each ordinal number $\alpha$ there is a distinct cardinal number ${\aleph}_{\alpha},$ and the ordinal numbers go on and on and on and $\ldots$ For ordinals less than ${\omega}^{\omega},$ see this MSE answer. For much larger countable ordinal numbers, see John Baez's Jun-Jul 2016 blog posts Large Countable Ordinals Part 1 & Part 2 (continued) – Dave L. Renfro Mar 16 '25 at 20:14
  • & Part 3. Finally, the ordinals continue into the uncountable realm. In a seemingly paradoxical way, it is even possible to have ${\aleph}{\alpha} = \alpha$ (equality as ordinal numbers) -- see the mathoverflow question [Intuition about ordinal fixed points $\alpha = \aleph\alpha$](https://mathoverflow.net/q/365052/15780) (this paradoxical situation is analogous to how the ordinal number ${\omega}^{\omega}$ is the ${\omega}^{\omega}$'th limit ordinal). – Dave L. Renfro Mar 16 '25 at 20:14
  • Basically $\aleph_{i+1}$ is the smallest ordinal size bigger than $\aleph_i$. – J.G. Mar 16 '25 at 22:52

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Cantor proved that the cardinality of $\mathbb{R}$ is greater than the cardinality of $\mathbb{N}$. His original argument was not the diagonal argument which you've probably seen.

For a long time after that, these were the only two cardinalities Cantor knew of. He suspected that $|\mathbb{R}|$ was the next greater cardinality after $|\mathbb{N}|$, his famous Continuum Hypothesis (CH). But he didn't even have a proof that there was a next greater cardinality. (Could it be like the rational numbers, where there is no number "right after" 0?)

Eventually, using his theory of ordinal numbers, he proved the existence of the sequence of alephs, $\aleph_0<\aleph_1<\ldots$, with no cardinalities in between. His continuum hypothesis took the form $2^{\aleph_0}=\aleph_1$. (But he didn't write this equation down until decades after he proved that $|\mathbb{R}|>|\mathbb{N}|$. And he never stated the generalized continuum hypothesis, GCH: $2^{\aleph_\alpha}=\aleph_{\alpha+1}$, for any ordinal $\alpha$.)

He was aware of cardinalities greater than all the $\aleph_n$'s with $n\in\mathbb{N}$. The smallest of these is $$\sum_{n\in\mathbb{N}}\aleph_n$$ It's the cardinality of a union of sets $X_n$, where $X_n$ has cardinality $\aleph_n$. It's denoted $\aleph_\omega$. But then of course you have $\aleph_{\omega+1}$, $\aleph_{\omega+2}$, ...

As for reading recommendations: Stillwell's Roads to Infinity is very good. Set Theory: An Open Introduction is free to download, and has more about cardinals. At a more advanced level, there is Frank Drake's Set Theory: An Introduction to Large Cardinals, Keith Devlin's The Joy of Sets, and Lorenz Halbeisen's Combinatorial Set Theory.

Finally, you might want to take a look at my Set Theory Jottings.

Michael Weiss
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