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I'm familiar with Aleph Null, Aleph 1, and Aleph 2 but are there greater [uncountable] infinities beyond (examples) Natural numbers, Real numbers, and the number of curves that can pass through a point? If so, what are some tangible examples of these cardinalities: Aleph 3, 4, ...

poetasis
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3 Answers3

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Yes. Just keep taking power sets of power sets ad infinitum. There is no largest cardinal.

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The next one after $\aleph_2$ is $\aleph_3$. There's also $c$, the cardinality of the reals. In fact $c=\aleph_\alpha$ for some $\alpha$, but nobody knows which one...

Regading your new request for "tangible examples": In fact $\aleph_0$ is the cardinality of the integers, which I gather is what you mean by a tangible example. But no, $\aleph_1$ is not the carinality of the reals.

So what are these aleph things? One can prove that given a cardinal there is a smallest larger cardinal. So $\aleph_1$ is by definition the smmallest uncountale cardinal, $\aleph_2$ is by definition the smallest cardinal larger than $\aleph_1$, etc. That's as "tangible" as it gets.

Finally, regarding "Natural numbers, Real numbers, and the number of curves that can pass through a point?": Although you didn't actually say so, this sounds like you think the cardinality of that set of curves is larger then the reals. This is not so. If a "curve" is the graph of a continuous function, then the cardinality of the set of all curves, passing through a given point or not, is just $c$.

David C. Ullrich
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  • I thought c and ℵ1 were the same because it is the cardinality of reals. I don't think the continuum hypothesis requires the axiom of choice either. – poetasis May 24 '18 at 18:08
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    @poetasis No. That statement is known as the continuum hypothesis, and is known to not be decidable from the usual axioms of set theory. Indeed, in a precise sense ZFC can't prove anything about the exact value of the continuum in terms of the $\aleph$-hierarchy except that it can't have countable cofinality. – Noah Schweber May 24 '18 at 18:08
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    @poetasis No. I've heard that there are well-known popular books that state that $c=\aleph_1$, but it's not so. At least not provably so; it's been proved that you can't prove $c=\aleph_1$ and you also can't prove $c\ne\aleph_1$. – David C. Ullrich May 24 '18 at 18:10
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    @poetasis "don't think CH requires AC" sounds like you're saying AC is not needed to prove CH. In fact you can't prove CH with or without AC. – David C. Ullrich May 24 '18 at 18:20
  • It is slightly more tangible to say that $\aleph_1$ is the cardinality of the set of isomorphism classes of total orders on $\mathbb N$ that are well-orders. (But still nothing that really engages the imagination). – hmakholm left over Monica May 24 '18 at 18:38
  • @HenningMakholm Yes. Curiously, it seems to me that what's "tangible" depends on who you are. Many people find $c$ perfectly "tangible", being the cardinality of the reals. But (or so it seems to me) things like $\aleph_n$ are perfectly "tangible" to a set theorist, while $c$ is the "intangible" one... – David C. Ullrich May 24 '18 at 18:44
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To conceptualize larger cardinals, you can use power sets, for sure. If you want something larger than the cardinality of curves passing through a point, consider subsets of the set curves passing through a point.

$$|\mathcal{P}(A)|>|A|$$

Another way of looking at this approach is to consider the set of functions from a set of one cardinality to, for example, the set $\{0,1\}$. This set is equivalent to the power set of your original set, and its cardinality is greater. In symbols: $$|\left\{f|f:A\to\{0,1\}\right\}|>|A|$$

G Tony Jacobs
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