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The concept is infinity is HUGELY fun.

For example: an "indescribable cardinal" is one so large you can't describe it. Roughly speaking.

"The point of indescribable cardinals is that they are characterized by some degree of uncharacterizability" - Kanamori and Magador

(cont)
That sounds paradoxical! But it's not.

Very roughly, the idea is that any property you can write down about an indescribable cardinal is already true for some smaller cardinal. So, you can't pick it out as the smallest one with some property you can describe.

(continued)
John von Neumann described the so-called "universe" - the collection of all sets - as the union of a list of increasingly large collections of sets V(0), V(1), V(2), etcetera. But "etcetera" must go on for a REALLY long time!

(continued)

en.wikipedia.org/wiki/Von_Neuma…
In particular, to get the universe as the union of sets V(k), we need to let k range over all possible infinities.

A cardinal k is indescribable if, roughly, any statement that's true in V(k) is already true in V(j) for some j < k.

(continued)
So, we can't describe an indescribable cardinal k by saying it's the smallest cardinal such that something is true in V(k).

And that's how we describe them! 😀

For details, see Wikipedia: en.wikipedia.org/wiki/Indescrib…

(continued)
We can't prove indescribable cardinals exist using the standard axioms of set theory (the ZFC axioms) - but that's already true of smaller cardinals like "inaccessible" ones.

In fact, indescribable cardinals are near the *bottom* of the ladder of large cardinals!

Fun, fun, fun.
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