User blog comment:Denis Maksudov/M-notation (new version)/@comment-24920136-20160621003904/@comment-27513631-20160624115316

There is; it's the same as \(\aleph_0\), but with an ordering to it. And, when talking about recursion directly, there are practically no cases where cardinality is relevant; you can almost always just use the admissible hierarchy instead.

So I'd say that if you're going to have the concept of 'amount' among infinite ordinals when doing recursion, you're probably going to be using different ordinals of the same cardinality.