I'm reading the Wikipedia definition of an inaccessible cardinal and I'm trying to understand it.
On Wikipedia, a (strongly) inaccessible cardinal $\kappa$ is defined in the following way:
- $\kappa$ is uncountable.
- $\kappa$ is not the sum of fewer than $\kappa$ cardinals smaller than $\kappa$.
- For all ordinals $\alpha < \kappa$, $2^\alpha$ is also strictly less than $\kappa$.
(1) and (3) are straightforward.
(2) confuses me in two ways: how do you define the sum of a multiset of cardinals, and also, what cardinals is it intended to rule out? (3), intuitively, seems like a much more difficult bar to clear. It's not clear to me intuitively which accessible cardinals clear (3) but are caught by (2).
Focusing for a moment on how they're defined:
Here's my best guess of how to do it.
Let $\alpha$ be a nonzero ordinal such that $|\alpha| < \kappa$. Let $f : \alpha \to \kappa$.
$$ \sum f \;\;\text{is defined as}\;\; \left|\bigcup_{x \in \alpha} \{x\} \times f(x) \right| $$
Intuitively, I'm thinking of $\alpha$ as like a color palette for a paintbrush, and painting the elements of the different cardinals different colors.
This definition seems like it would be kind of hard to work with though and it doesn't generalize to, for example, infinite products.
Is there a better way to define the sum of a set of cardinals?
If $(\kappa_i)_{i < \lambda}$ is a family of cardinals, the sum is usually defined to be $\left\vert \bigcup_{i < \lambda} (\kappa_i \times \{i\}) \right\vert$. In fact, the specific sets used aren't important; if $(A_i)_{i < \lambda}$ is any family of pairwise disjoint sets and $|A_i| = \kappa_i$ for all $i$, then $\sum_{i < \lambda}\kappa_i = \left\vert\bigcup_{i < \lambda}A_i\right\vert$.
For your other question, perhaps the easiest example is this: Suppose that the Generalized Continuum Hypothesis holds, so $2^\kappa =\kappa^+$ for all cardinals $\kappa$. Then $\aleph_\omega$ satisfies (1) and (3), but not (2), because $\aleph_\omega = \sum_{i < \omega}\aleph_i$.