I don’t quite buy this, because we want to abstract over the stuff reachable by a particular generic process, say, over the natural numbers. We want to say, in mathematical language (i.e. as formulas), things like “whatever x we can reach in this kind of process, property P will hold for it”. This requires abstracting over all possible x, and in that abstraction, “all possible x” becomes itself an object. And we call those kinds of objects sets.