User blog comment:Edwin Shade/How Do I Evaluate BEAF Arrays In Two Dimensions ?/@comment-30754445-20170827183955/@comment-5529393-20170830233436

@Googleaarex: Again, we need to specify precisely what it means to "construct with smaller elements". Every ordinal is precisely the set of ordinals smaller than itself, so in a certain sense every ordinal is constructible from smaller elements.

But, one might want to add the extra condition that the ordinal must be constructible from a smaller number of ordinals than the ordinal itself. For a successor ordinal $$\alpha + 1$$, I think it is reasonable to say that it is constructible from smaller ordinals, since it is just the next ordinal after $$\alpha$$. So we can restrict our attention to limit ordinals. So, we can parse the question as, "for which limit ordinals $$\alpha$$ can we construct a set of ordinals smaller than $$\alpha$$ such that the set has order type less than $$\alpha$$".

In fancy math terminology: we define the 'cofinality of an ordinal $$\alpha$$ to be the smallest order type of any subset of $$\alpha$$ whose supremum (least upper bound) is $$\alpha$$. We define an ordinal to be regular if its cofinality is itself, and singular otherwise. So the question becomes, which ordinals are regular, and which are singular?

One can prove without too much difficulty that the cofinality of an ordinal is in fact a cardinal. (A cardinal is an ordinal that is the smallest ordinal of a particular cardinality. For example, the countable ordinals are those ordinals that are at least $$\omega$$ and less than $$\omega_1$$.  The smallest ordinal in this interval is $$\omega$$, so it is the unique cardinal in that interval.) So for an ordinal to have a cofinality of itself, it must be a cardinal. Now, the infinite cardinals are enumerated by the $$\aleph$$ function, so every infinite cardinal is of the form $$\aleph_\alpha$$ for some ordinal $$\alpha$$ When $$\alpha$$ is a successor ordinal, $$\aleph_\alpha$$ is called a successor cardinal, and when $$\alpha$$ is a limit ordinal, $$\aleph_\alpha$$ is called a limit cardinal. It turns out that every successor cardinal is a regular ordinal. (Note: This statement requires the axiom of choice. Without the axiom of choice, it is even possible that every successor cardinal has cofinality $$\omega$$!)  We can prove this by proving that, for any $$\alpha$$, a collection of $$\aleph_\alpha$$ sets of size at most $$\aleph_\alpha$$ can have cardinality at most $$\aleph_\alpha$$; thus, $$\aleph_{\alpha+1}$$ cannot be the supremum of $$\aleph_\alpha$$ or fewer cardinals. On the other hand, most limit cardinals are singular; for example, $$\aleph_\omega$$ is the supremum of $$\aleph_n$$ for n finite, hence $$\aleph_\omega$$ has cofinality $$\aleph_0$$. In fact, it is consistent with ZFC that all limit cardinals are singular. But it is also consistent with ZFC that some limit cardinals are regular; these cardinals are known as weakly inaccessible cardinals.

@Denis: I too have seen "countable" defined this way. I have even seen people use both "countable" and "denumerable", where one of those terms includes the finite ordinals and the other does not. However, I believe most of the time "countable" is defined to include finite ordinals.