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I've seen the following two definitions in my slides:

  1. $S \subseteq \aleph$ is semi-decidable iff there exists a partially computable function g where $S = \{x \in \aleph\ |\ g(x)\downarrow \}$
  2. $S \subseteq \aleph$ 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?

Clever7-
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    Surely it's a meant to denote the set of natural numbers. This is not standard notation - maybe it's just a typo for $\mathbb{N}$? – Alex Kruckman Mar 30 '24 at 13:26
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    If $\aleph$ is to be used by itself, it should be defined first. (Or maybe it is a misprint for $\mathbb N$ ?) – GEdgar Mar 30 '24 at 13:26
  • I continue to see it in my class slides though, and I haven't seen a definition. It could be a placeholder for a set of sets of some kind, but I guess then you'd just use the universal set. – Clever7- Mar 30 '24 at 14:23

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