I've seen the following two definitions in my slides:
- $S \subseteq \aleph$ is semi-decidable iff there exists a partially computable function g where $S = \{x \in \aleph\ |\ g(x)\downarrow \}$
- $S \subseteq \aleph$ is re iff $S = \phi$ or there exists a totally computable function $f$ where $S = \{y\ |\ \exists x f(x) == y\}$
I know that $\aleph_0$ or $\aleph_1$ represent set cardinalities but what does it mean if there is no subscript and it is used in the above context?