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

Let $A\subset N$ 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.