# Can the concept of antisets be used for a neo-mechanist causal set theory?

Background information:

1. "Causal Approaches to Scientific Explanation," sec. 1. My takeaway here is looking at individual existential quantifications, i.e. quantifying over individual causal elements: the relevant kind of ordering (for the encompassing sets) is read off and up from specified elements.
2. "Causal sets". An overview of the program. I don't know if it counts as normal science, fringe science, or meta-science; I'm working with it at least for the sake of "toy models" in physics.
3. Set theory with antielements? (MathSE question) I quote here from Noah Schweber's reply:

... ∪ isn't a primitive operation, it's defined in terms of ∈, and the logical associativity of ∨ directly leads to the algebraic associativity of ∪ ...

I interpret this as indicating that antisets are conceptually stable only in a sort of tensed/dynamical theory; in a timeless state, we would have x, its antielement y, the union operation (in this case, static fact) of their absorption into {x, y}, and yet then if xy = 0 = {}, we end up with {x, y} = {}, which is a contradiction. (More longwinded: we'd have x ∈ {x, y}, y ∈ {x, y}, and yet also that neither of these held "after" the element/antielement pair canceled itself out.) We could switch out the equals sign for a right arrow, granted, but again, this should be in something like a tensed logic, or rather the arrow notation thematically indicates a dynamical transition anyway.ω

As far as I can tell, if we install a "clock" on the interval notation, we can instead represent a noncontradictory equivalence of the (x, y)-doubleton and the empty outcome: {x, y}t = {}t+1. I haven't actually thought through that possibility much because my mind pretty quickly was diverted to the following concept:

1. Absorbing and repelling sets: sets with elements x such that, whenever they start out lacking elements y, then they gain those elements whenever the x are added; sets with elements x such that, whenever they start out having elements y, then they lose those elements whenever the x are added.

Then we could go on to "elements that, when repelled, are repelled into the set-theoretic 'ether' or which (necessarily or not) end up in some other specific sets," etc.

Suppose, then, one made toy models of physical charge relations (not the matter-antimatter relation directly, not even the electrical-charge function, but just a charge function "in general") in terms of sets like the above. Since we are not (clearly) talking about united all-encompassing laws imposing order "from the top down" but simply indicating a fairly specific kind of dynamical set relation, are we working with a causal set theory that is also neo-mechanist in character? (Note that we are not claiming the nonexistence of top-down laws, and in fact one might style the type of antielement dynamics as a top-down law. But if I wrote out the intended quantifiers for the above, I would be mostly using the existential quantifier on individual, relatively local variables, rather than the universal quantifier in tandem with conditional arrows.)

ωSuppose, for example, that antielements never run out in the well-founded transet. If we take the universal transet over all element-antielement pairs, the outcome would be the equivalent of a set both completely empty and completely universal, or a multiset with |ORD|-many copies of the empty set inside it. One might take this for a novel illustration of the set-theoretic multiverse (i.e. "there is a concept of sets in terms of element/antielement pairs that cannot be composed into a one individual stable universe"), or one could take it to show that the concept of antisets goes beyond a theory of timeless pure sets by default, or one might think of probably a bunch of other things I haven't myself thought of yet.