I've been doing a bit more "stir in fancy set theory" over the weekend, and believe I have an improved recipe! This builds on the idea to treat chains and loops as a single "entity" and look for a cause of that entity. It is a lot subtler than just throwing every entity together into one super-duper-entity.
Here are a bunch of premises that I think will do the trick:
A1. The collection of all entities is a set E, with two relations C and P on E, such that: x C y if and only if x is a cause of y; x P y if and only if x is a part of y.
A2. The set E can be well-ordered
Note: This ensures we can apply Zorn's Lemma when considering chains in E, but is not as strong as the full Axiom of Choice. If the set E is finite or countable, for instance, then A2 applies automatically.
A3. If x C y and x P z then z C y.
Informally, "anything caused by a part is caused by the whole".
Definitions: We define <= such that x <= y if and only if x = y or there are finitely many entities x1, ..., xn such that x1 = x, xn = y and xi is a cause of xi+1 for i=1.. n-1. Say that a set S is a "chain" in E iff for any x, y in S we have x <= y or y <= x. Say that such an S is an "endless chain" iff for any x in S there is some y not equal to x in S with y <= x. Say that an entity y is "uncaused" if and only if there is no z distinct from y with z <= y. Also say that x is a "proper part" of y iff x is not equal to y but x P y.
Note: These definitions ensure that <= is a pre-order on E. Note that an endless chain may be an infinite chain of distinct elements, or a causal loop.
A4. Let S be any endless chain in E. Then there is some z in E such that every x in S is a proper part of z.
Lemma 1: For any chain S in E, there is an element x of E with x <= y for every y in S.
Proof: Suppose S has an end (not endless). Then there is some x in S such that for no other y in S is y <= x. By the chain property we must have x <= y for every member y of S. Alternatively, suppose that S is endless, then by A4, there is some z in E such that every x in S is a part of z. Now consider any y in S. There is some x not equal to y in S with x <= y, so there are x = x1... xn = y with each xi C xi+1 for i=1..n-1. Further, by A3, as x C x2, we have z C x2 and hence z <= y.
Lemma 2: For any x in E, there is some y in E such that: y <= x, and for any z <= y, y <= z.
Proof: This is the version of Zorn's Lemma applied to pre-orders.
Theorem 3: For any x in E, there is some uncaused y in E such that y <= x.
Proof: Take a y as given by Lemma 2 and consider the set S = {s: s <= y}. By Lemma 2, y <= s for every member of S, and if S has more than one element, then S is an endless chain. So by A4 there is some z of which every s in S is a proper part, which implies that z is not in S. But by the proof of Lemma 1, z <= y, which implies z is in S: a contradiction. So it follows that S = {y}, which completes the proof.
I've also got some premises for aggregating multiple uncaused entities into a single entity. This gives another approach to "uniqueness". More on my next comment, if you're interested.
For uniqueness, we build on the idea of all uncaused causes being part of a whole. The following premises look interesting here:
B1. If x P y and y P z then x P z; x = y if and only if x P y and y P x.
This states that P is a partial order, which is reasonable for the "part of" relation.
B2. If S is any chain of parts, such that for any x, y in S we have x P y or y P x, then there is some z in E of which all members of S are parts.
This states that E is inductively ordered by the "part of" relation.
B3. If x C z and y P z then x C y.
In...
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