Intuition regarding: $\kappa^{+}=|\{\kappa\leq\alpha\lt \kappa^{+}\}|$

Question

This question is motivated by Cor. 3.7.9 in Devlin's "Joy of Sets."

The corollary says: For $\kappa\geq \aleph_0$

$\kappa^{+}=|\{\kappa\leq\alpha\lt \kappa^{+}\}|$

i.e., the set of all ordinals of cardinality $\kappa$ has cardinality $\kappa^{+}$.

Does this mean that there are $\aleph_4$ ordinals ($\alpha$'s) all with the same cardinality $\aleph_3$ with $\aleph_3\leq\alpha\lt \aleph_4$. (If it is correct that $\kappa^{+}=\aleph_4$ means $\kappa=\aleph_3$.)

What do they look like? (The foibles of a self-studier.)

Thanks

Answer 1

Building off of Asaf's answer (again!):

You also ask, re: ordinals between $\kappa$ and $\kappa^+$ (say, $\kappa=\aleph_3$),

What do they look like?

Well, the first few look like this: $$\omega_3, \omega_3+1,\omega_3+2, ..., \omega_3\cdot 2, ..., \omega_3^2, ... \omega_3^{\omega}, ..., \omega_3^{\omega_3}, ...$$

Here $\omega_3$ is the smallest ordinal of cardinality $\aleph_3$; strictly speaking, $\omega_3$ and $\aleph_3$ are two symbols denoting the same set, but it's useful to distinguish their contexts (especially because ordinal and cardinal arithmetic are so different, both in terms of results (cardinal arithmetic is commutative, ordinal arithmetic isn't) and in terms of definitions (ordinal exponentiation is a very different thing from cardinal exponentiation)).

You should notice someting familiar here: this looks a lost like what we see when we try to write down some countable ordinals. And, just like there, we rapidly run out of easy-to-describe ordinals of the appropriate type; this is because any "reasonable" way of describing ordinals, called a "notation system," has "small fixed points" - that is, ordinals which we care about (e.g. are smaller than $\kappa^+$) but which can't be described in a nontrivial way using that notation system. The motivating example here is the ordinal $\epsilon_0$ - this is the least ordinal $\alpha$ satisfying $\omega^\alpha=\alpha$, and is often suggestively written as $$\large\epsilon_0=\omega^{\omega^{\omega^{\omega^{{}^{.^{\,\,.^{\,\,.}}}}}}}$$ - in the context of the notation system of Cantor normal form.

The difficulty of describing large countable ordinals is discussed at this math.stackexchange question, and is I think a good starting point to understanding why it's so hard to get a "complete description" of e.g. the ordinals between $\omega_3$ and $\omega_4$.

Answer 2

Yes. That is exactly what it means.

One easy way to see this is to note that $\{\alpha\mid\kappa\leq\alpha<\kappa^+\}$ is exactly the set $\kappa^+\setminus\kappa$. So $\kappa^+=(\kappa^+\setminus\kappa)\cup\kappa$, as a disjoint union it means that the cardinality $\kappa^+$ is the sum of the two cardinals.

But $\kappa^+=\lambda+\kappa$ means necessarily that $\lambda=\kappa^+$.

(Another way, which is essentially the same, is noting that the ordinal sum $\kappa+\kappa$ is equipotent with $\kappa$, and repeating the essentially the same argument as above to get that the set $\kappa^+\setminus\kappa$ has size $\kappa^+$.)