I’m reading Peter Hallward’s book on Badiou, _A Subject to Truth_. I find the Appendix helpful as an overview of the math stuff. Here’s one thing I don’t understand. Hallward lists the axioms basic to Zermelo- Fraenkel set theory, the first two of which are

“1. Axiom of extensionality: “If two sets have the same elements, then they are identical; in other words, a set’s identity is determined entirely by its elements.

2. Null-set axiom: “There is a set that contains no elements, written Ø.” This is “the only set whose existence is directly asserted. Every other set is constructed in some way or other from this set.”

Fine and good. Here’s where I don’t get it. Hallward adds parenthetically under axiom 2 “Given the axiom of extensionality, the uniqueness of the empty set follows as an obvious theorem.” Why is this obvious?

It seems to be something like this: for any set X, X contains Y elements. If a set N contains the same Y elements as X then N=X. If set X is the set “cats who live in my apartment,” set N is the set “living cats belonging to my wife,” and set M is “cats I have met which I have known to be older than 19 years of age” then in all three cases Y=1 and that 1 is the same cat. Therefore M=N=X where the equal sign signifies identity. Unless I’ve made an error, this seems to be how things are working, and to be an unproblematic use of axiom 1.

This is distinct from something like the following where set X is “cats who live in my apartment,” set N is “my living grandmothers” and set M is “the apartments I live in.” In this case for all three sets Y=1, which means that these are quantitatively the same but M≠N≠X because the single element contained in each is not the same single element: my cat is not my grandmother is not my apartment. The sets in this second case could be said to be equal but are not identical.

Turning to axiom 2, axiom 2 says there is a set with no elements, the empty set or null set. If X is the empty set then X contains Y elements such that Y=0. If set X is “empty set”, set N is “null set”, and set M is “set with no elements” then Y=0. Fair enough. But why is this a case of these sets being identical instead of equal? In all three Y=0, but is that 0 is the same no elements in all three? How is a non-element comparable such that it can held as either the same as or different from another non-element? Put differently, on what basis are there non-elements vs a single non-element? The latter seems to be implied in this case if M=N=X where Y=0 and that 0 is the same. This implies a single zero or single emptiness. Why not many zeroes or many emptinesses (which is to say, many empty sets rather than one empty set)? That (many) seems no less (and no more) coherent to me than one.

Is there a difference here if we keep X as empty set but make N “the set of round squares” and M “my currently living grandfathers”? As far as I can tell, according to the view that there’s one empty set, these would be the same. In all three cases Y=0 and it’s the same zero. But while I admit this is a bit silly, I don’t see why my no currently living grandfathers should be the same as my no round squares. That is, I don’t see why their sameness is any more obvious than their difference. It may end up that neither is coherent, such that 0 is neither the same as nor different from 0.

Maybe that’s the question I guess. In terms not of equality but in terms of identity, does 0=0? Does 0≠0? Neither really makes sense to me (it seems like a bad pair of questions). Hallwards claim that the uniqueness of empty set seems to involve a slip from “=” as equality to “=” as identity.

Of course, I may be totally off base, this is all new to me and I feel clumsy at it. Can anyone sort me out?