No more than half of anti-sets are big

Background on anti-set theory here.

Suppose that somebody tells you that they have in their hands a model of anti-set theory containing 50 anti-sets. The axiom of anti-infinity gives us some nice immediate restrictions on how large these anti-sets can be. As a quick reminder, the axiom of anti-infinity tells us that every anti-set X must contain another anti-set Y that has a successor that’s not in X. The successor of X is defined to be X ⋃ {X} (an anti-set containing everything in X plus X itself). One big difference between successors in ZFC and successors in anti-set-theory is that anti-sets aren’t always guaranteed to have successors. Another is that anti-sets can be their own successors (this happens if the anti-set contains itself).

Ok, so anti-infinity immediately tells us that there can be no universal set (i.e. a set containing the entire universe). Since every set X must contain a set that has a successor outside X, no set can contain everything, or else it would contain all the successors!

So there are no anti-sets of size 50. How about anti-sets of size 49? It’s recently been proved that there can only be at most 25 sets of size 49. And in general, in a universe of size N no more than half of the anti-sets can be size N-1. Let’s prove it!

First, a naming convention: in a universe of size N, let’s call any sets of size N-1 big sets, and any other sets small sets. Big sets are as big as an anti-set can be. To satisfy anti-infinity, any set X must contain an element Y that has a successor that’s not in X. We’ll call Y the anti-infinity-satisfier for X.

Now, some observations. Suppose that a big set X has a successor. This successor is either size N-1 (if X contains itself) or size N (if not). But no set is size N! So any big set that has a successor, must be its own successor. But this means that big sets cannot be anti-infinity-satisfiers. If a big set X was an anti-infinity-satisfier for some other set Y, then its successor could not be in Y. But its successor is X, and X is in Y!

What this means is that every set must contain a small set as its anti-infinity-satisfier. Now, consider any big set X. X must contain some small set z that serves as an anti-infinity-satisfier for it. But for z to serve as an anti-infinity-satisfier, z must have a successor that’s not in X. What this means is that X must be missing z’s successor.

In other words, every big set must be missing at least one small set’s successor. But remember, big sets are size N-1. This means that they contain every set in the universe besides one. So now we know that the “missing set” that characterizes each big set is always a successor of a small set!

Let’s say that our universe contains k small sets. Then there are at most k successors of small sets. Any big set must be the set of all sets besides one of these successors. Thus by extensionality, there are at most k big sets. Therefore there can never be more big sets than small sets!

And that’s the result: In any finite universe, at most half of the anti-sets are big.

This resolves a conjecture that I posted earlier as an open question: in an N-element model, are there always more than three small sets? The answer is yes. If we had fewer than three small sets, then we’d have to also have fewer than three big sets. This means that our universe is at most size four (two small sets and two big sets). But no four-element models exist! QED.