# Computability and Logic

This page contains scribbles and random musings as I process the contents of *Computability and Logic*. It’s not really for public consumption (unless you happen to care about my specific confusions).

Something that bugs me is that there is a classification of (partial) recursive(ly enumerable) sets, functions, etc. but textbooks seem to only state *some* of these results, rather than presenting a table with *all* the results visible at once.

Let \(f : \mathbf N \to \mathbf N\) be a partial or total function. If the {domain / range} of \(f\) is {recursively enumerable / enumerable in increasing order} then \(f\) is ____.

Let \(A \subset \mathbf N\) be a {recursive / recursively enumerable / recursively enumerable in increasing order} set. Then \(A\) can be the {domain / range} of a {recursive / partial recursive / primitive recursive} function.

Also there are multiple ways to pass between functions and sets here.

- One way to go from a set to a function is to take the characteristic function of the set.
- Another way to go from a set to a function is to make a function enumerate (possibly in increasing order) the set.
- Another way to go from a set to a function is to make the set the domain or range.
- Given a (partial) function, we can take the domain or range to get a set.
- We can define some relation, then quantify over some subset of the variables until we get a dimension we want, then define a set using that relation.
- Set theoretically, a function
*is*a set. But I don’t think you can say anything interesting about the set that the function happens to be.