# Is it possible to define powersets without first defining the subset relation?

This ties back to a previous question I asked, which was about the notion of "to define $$X$$, you must first define $$Y$$". The current question is about a specific application of that notion. Is it possible to define powersets without first defining the subset relation? To make my question more precise, suppose we are working in $$ZFC$$ set theory. Does defining the binary relation, "$$X$$ is the powerset of $$Y$$" require one to first define the relation "$$X$$ is a subset of $$Y$$"? In other words, we are really making a definitional extension of $$ZFC$$. Of course, I know that one can directly make the definitional extension by defining "$$X$$ is the powerset of $$Y$$" in terms of the primitive notion of $$\in$$, but my question is, in that definitional extension, must you always use, either explicitly or implicitly, the binary relation of subset?

• I think this is currently too vague to admit an answer. In particular, since $\subseteq$ can be defined from $\mathcal{P}$ and $\in$ as $$A\subseteq B\iff A\in \mathcal{P}(B),$$ I think it will be very hard to identify a precise sense in which we can get to $\mathcal{P}$ without "in effect" going through $\subseteq$. Jan 27 at 21:03
• Like @NoahSchweber, I think your question is too unclear to have a definite answer. Perhaps one could firm it up by asking if one can axiomatise ZFC using the relation "$X$ is the power set of $Y$" as the only primitive. My guess is not, but I don't see how to prove that just now. Jan 27 at 22:04
• @RobArthan Powerset alone has very little power (hehehe): it just describes a total injective function on the universe whose image misses infinitely many things. That gives a decidable theory! Jan 27 at 22:07