I’m relocating part of a thread that was originally on “Welcome to Less Wrong” but has wandered way off topic. It also seems that a remote ancestor comment was heavily downvoted, discouraging further contributions in the original place. So I’m moving into the Open thread.
(Huh. One of the ancestors to this comment—several levels up—has been downvoted enough to require a karma penalty. I wonder if there should be some statute of limitations on that; whether, say, ten levels of positive-karma posts can protect against a higher-level negative-karma post?)
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.
Definitions: We define a relation C such that x C y iff there are entities v, w such that v P x, y P w and v C w.
Note: This gives a broader causal relation which automatically satisfies “if x C y and x P z then z C y” as well as “if x C z and y P z then x C y”, loosely “anything which is caused by a part is caused by the whole” and “anything which causes the whole, causes the part”. So we don’t need to state those as extra premises.
We then define a further relation ⇐ such that x ⇐ y iff x = y, or there are finitely many entities x1, …, xn such that x1 = x, xn = y and xi C* xi+1 for i=1.. n-1.
Note: This construction ensures that ⇐ is a pre-order on E.
Say that a subset S of E is a “chain” iff for any x, y in S we have x ⇐ y or y ⇐ x. Say that S is an “endless chain” iff for any x in S there is some y in S distinct from x with y ⇐ x. We shall say that y is “uncaused” if and only if there is no z in E distinct from y with z C* y (this of course implies there is no z distinct from y with z C y, but it also implies that y isn’t part of anything which is caused by something distinct from y). Say that x is a proper part of y iff x is distinct from y but x P y.
A3. 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 entity x in E such that x ⇐ y for every y in S.
Proof: Suppose S is 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 A3, there is some z in E of which every x in S is a part. Now consider any y in S. There is some x not equal to y in S with x ⇐ y, so there are entities x = x1… xn = y with each xi C xi+1 for i=1..n-1. Further, as x P z 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 every z ⇐ y, we must have y ⇐ z.
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 A3 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.
We now partition E into three subsets. I are the “inert” entities, which do not cause anything and have no causes themselves. (Note that the new version allows there to be some of these, unlike the previous version; you can think of them as abstract entities like numbers, sets, propositions and so on, if you want to). Formally I = {x in E: there is no y distinct from x with x C y or y C x}. U are the “uncaused causes”—formally U = {x in E: there is no y distinct from x with y C x, but there is z distinct from x with x C z}. O are all the “other”, caused entities, so that formally O = {x in E: there is some y distinct from x with y C* x}.
B1. If S is any subset of U such that for any x, y in S we have x P y or y P x, (call such an S a “chain of parts”), then there is some entity z of which all members of S are parts.
B2. Suppose that y ⇐ x and z ⇐ x. Then there is some entity w such that: w ⇐ x; w ⇐ y or y P w; w ⇐ z or z P w.
(EDIT: Restated to ensure that Theorem 4 properly follows.) Informally, the idea is that y and z can’t independently cause x without any further causal explanation. So there must be some common cause, however each of them may be part of that common cause.
Definition: Say that entities x and y are causally-connected if and only if x=y or there are finitely-many entities x=x1,..,xn=y with each xi C xi+1 or xi+1 C xi for i=1..n-1.
B3. Any two entities x, y in O are causally-connected.
Informally, O doesn’t “come apart” into disconnected components, such as a bunch of isolated universes. Premises B1-B3 turn out to be necessary for Theorem 6 to hold, as well as sufficient (see below). So they can’t be made any weaker!
Theorem 4: For any x in O, there is a unique entity f(x) in U such that: f(x) ⇐ x, and any other y in U with y ⇐ x satisfies y P f(x).
Proof: For any x in O, define a subset U(x) = {y in U: y ⇐ x}; this is non-empty by Theorem 3. Consider any chain of parts S that is a subset of U(x). If it has at least two members, then by B1 there is some z in E of which all members of S are parts, and such a z must be in U. (If not, then note any w C z would also satisfy w C s for each member s of S, which would require them all to be equal to w). Also since y ⇐ x for any member of S and y P z we have z ⇐ x. So z is also a member of U(x). Or if S is a singleton—say {z} - then clearly all members of S are parts of z, and z is also in U(x). By application of Zorn’s Lemma to U(x), there is a P-maximal element f(x) in U(x) such that there is no other y in U(x) with f(x) P y. By B2, for any other y in U(x) there must be some z in U(x) with f(x) P z and y P z; given f(x) is maximal we have z = f(x) and so y P f(x). This makes f(x) the unique maximal element of U(x).
Theorem 5: For any x, y in O, f(x) = f(y) if and only if x and y are causally-connected.
Proof: It is clear that if f(x) = f(y) then x is causally-connected to y (just build a path backwards from x to f(x) and then forward again to y). Conversely, consider any two x, y in O:
a) If x C y, then f(x) is in U and satisfies f(x) ⇐ y so we have f(x) P f(y). Since x is not f(x), we have f(x) C x2 ⇐ x for some x2, and hence f(y) C* x2 ⇐ x i.e. f(y) ⇐ x which means f(y) P f(x), and so f(x) = f(y).
b) If z is in U with z C x and z C y, then z P f(x) so f(x) ⇐ y and f(x) P f(y); similarly, f(y) P f(x) so that f(x) = f(y).
The result now follows by induction on the length of the causal path connecting x to y.
Theorem 6: If O is non-empty, then there is a single entity g in U such that: f(x) = g for every x in O, and y P g for every y in U.
Proof: Assuming O is non-empty, take any element y in O, and set g = f(y); then the result that f(x) = g for any x in O follows from Theorem 5 and B3; further, for any y in U, there is some x in O with y C* x, so by Theorem 4, y P f(x). If there are no elements of O (meaning there are none in U either) then the Theorem is trivial.
Finally, note that B1, B2 and B3 are entailed by the statement of Theorem 6. For B1, we can just take g as the relevant z. For B2, we can take g as the relevant w. B3 follows using using the first part of Theorem 5 (just track from x back to g, then forward to y again).
I’m just about done now, so unless there are errors in the above proof will leave it. What are the residual weak points? Well, B2 and B3 have been weakened a bit, but are still basically unjustifiable (we can imagine them being false without absurdity) and the above re-work shows they are needed for the uniqueness conclusion (Theorem 6). Also, we have the weakness of not deriving anything else useful about g.
Note: This gives a broader causal relation which automatically satisfies “if x C y and x P z then z C y” as well as “if x C z and y P z then x C y”, loosely “anything which is caused by a part is caused by the whole” and “anything which causes the whole, causes the part”. So we don’t need to state those as extra premises.
This will lead to a problem.
Consider assumption A3:
A3. 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.
Consider any endless chain consisting of at minimum two elements. Consider two elements in that chain, x and y, such that x C y. x and y are both proper parts of z. Therefore, x C z, and z C y. But then we have a longer chain; using x C z C y in place of x C y. Each element of that longer chain must then, by A3, be a proper part of a larger entity, z2. But then, similarly, we can construct the chain using x C z C z2 C* y. There are therefore an infinite number of entities z, z2, z3, z4… and so on, each including the one before it as a proper part (and nothing that is not part of the one before it).
Furthermore, anything which is a proper part of anything else is a part of such an infinitely recursive loop by default.
Consider any endless chain consisting of at minimum two elements. Consider two elements in that chain, x and y, such that x C y. x and y are both proper parts of z. Therefore, x C z, and z C* y.
It follows that z C y but it does not follow that x C z or that y C z.. The “whole” z may be a cause of its parts, without in turn being caused by its parts. Note that by construction of C it is true that if x is a cause of y and x is a part of z, then z C y. However, it is not generally true that if x is a cause of y and z is a part of x then z C y.
As an example of the intuition behind this: suppose I have a thermostat box containing two circuit boards. Board 1 is connected into my home heating system; Board 2 is a spare not connected into anything. It is true that Board 1 causes my heating to come on. It is true that the thermostat (of which Board 1 is part) causes my heating to come on. It is false that Board 2 (which is part of the thermostat) causes my heating to come on.
But then we have a longer chain; using x C z C y in place of x C* y.
You are right that when adding z, we now get a longer chain {x, y, z}, but this won’t in general be an “endless chain” (the new z may well be an end).
Consider any endless chain consisting of at minimum two elements. Consider two elements in that chain, x and y, such that x C y. x and y are both proper parts of z. Therefore, x C z, and z C* y.
It follows that z C y but it does not follow that x C z or that y C* z.
No, you need x C y and z P y to get x C z (be careful about which way the P relation is going).
The intuition is “Anything which is a cause of the whole is a cause of the part”, not “Anything which is a cause of the part is a cause of the whole”. Again, there are intuitive examples here. (Compare me baking a cake for a child’s birthday party vs just buying the cake from a shop, and putting a few sprinkles and candles on the top. In the second case, I am a cause of some part of the cake as presented to the child, but not the whole cake, and if someone says “Wow that cake tasted delicious!” I’d have to admit I didn’t make it, only decorated it).
I make a cake. I am a cause of the cake. The cake contains eggs. I am not the cause of the eggs. I think “what causes the whole, causes each part” is a bad intuition to have.
In general, I think it is an error, and a source of confusion, to think of things rather than events having, or being, causes. I know people sometimes do, and I’ve gone along with it in #1 above, but I think it’s a mistake.
Why would anyone assume A3? It seems really arbitrary. Exception: you might believe A3 because you believe in an entity of which all others are parts. See below.
If E includes an entity V of which all others are parts (call it “the universe”) then, provided C is reflexive, V C* anything-you-like. And I think it’ll then turn out that the way all your theorems work is that V is the canonical uncaused cause of everything. Which is a bit dull and wouldn’t satisfy many theists. Perhaps something more interesting happens if you make C irreflexive instead, so that things don’t count as causes of themselves.
Fair points, though there is in fact a lot of disagreement about what are the basic relata of the causal relation: see the SEP entry for example. When we apply causation to entities (which we can sometimes do, as in your example) then “A causes B” probably means something like “at least one event in which A is involved is a cause of every event in which B is involved”.
On counterexamples to “what causes the whole, causes the part” : possibly an even stronger counterexample considers just one of the atoms in the cake. However, we must be careful here: it is only some temporal part of the egg (or of the atom) which is part of the cake; the eggs/atoms in their full temporal entirety are NOT parts of the cake in its full temporal entirety. We could perhaps treat the relevant temporal part (“egg mixed into cake” or “atom within cake”) as an “entity” in its own right, but then it does seem that by making the cake, I am a cause of all the events which involve that particular “entity” (since I put the egg/atom into the cake in the first place).
In any case, note that the most recent version of the argument doesn’t actually need to assume this “cause-whole ⇒ cause-part” applies to C, since it only ever uses the constructed relation C instead. The conclusion is still interesting, since if nothing Cs the entity g, then nothing Cs it either, and if g causes some whole of which each entity is a part, that is still an interesting property of g. The argument makes no assumptions on whether C is reflexive or not.
On A3, I’m not totally sure of the circumstances in which we can aggregate entities together and treat them as parts of a single entity, but if the entities are causally related (and particularly if they are causally-related in an odd way, like an endless chain), then it does make some sort of sense to do this aggregation. After all, we immediately want to ask the question “How could there be an endless chain?” a question which does treat the “chain” as some sort of an entity to be explained. If entities are not causally related (they are in different universes), lumping them together seems much less natural.
Finally, on the “maximal entity” approach, CCC I believe discussed this in the original thread before I lifted here, and he seems to find it theologically interesting.
Looking over the “recipe” again, I notice one thing I failed to notice earlier; at no point have you formally defined the uncaused cause as God. That’s a weak step, but if you actually want to use that argument to argue for monotheism (instead of merely the presence of an uncaused entity, nature unknown) then I think it’s a necessary step. Some of the assumptions necessary to eliminate multiple uncaused causes are a bit weak, but I think that you have a very good argument that somewhere in the history of the universe there must be at least one uncaused cause of some sort.
No, I omitted that step for reasons discussed in the earlier thread: this gives too weak a “God” to be any interest to anyone, and is downright confusing.
The only way I can think to get back to some form of traditional theism is to add a premise saying that “every entity not of type G has a cause” (insert your favourite G) and then perhaps to pull the modal trick of claiming all the premises are possible...
I have spotted an error in the statement (and proof) of Theorem 5, and then Corollary 6. The issue is that for any uncaused y we must have f(y) = y, so if there are several uncaused entities then they can’t all have f(y) = g. The revised statements should go like this:
Theorem 5: Let x and y both have causes. Then f(x) = f(y) if and only if x and y are causally connected.
Proof: It is clear that if f(x) = f(y) then x is causally-connected to y (just build a path backwards from x to f(x) and then forward again to y). Conversely, suppose that x C y, then f(x) is uncaused and satisfies f(x) ⇐ y so we have f(x) P f(y); since x is caused, there are f(x)=x1,...,xn=x such that each xi C xi+1 for i=1..n-1, then by A3 we have f(y) C x2 and hence f(y) ⇐ x, which implies f(y) P f(x) and so by B1 f(x) = f(y). Next, suppose that for some uncaused z we have z C x and z C y; then z P f(x) which implies by A3 that f(x) C y and hence f(x) P f(y); similarly, f(y) P f(x) so by B1 f(x) = f(y). By an induction on the length of any other path connecting x to y, we have that f(x) = f(y).
Corollary 6: There is a single g in E such that: f(x) = g for every x in E with a cause, and every uncaused y P g.
Proof: Suppose there is at least one entity x with a cause, then set g = f(x). For any other caused entity y, f(y) = g by Theorem 5 and B5, and for an uncaused y, B5 implies y ⇐ x, so that y P g. Lastly, if there are no caused entities, then B5 implies that E = {y} for some uncaused y, so we can just pick g = y.
I have also spotted a way of weakening or removing some of the premises (in particular A3, and B1 to B4). I will update with that later today.
I’ve had another look at the argument, and spotted a way to remove the step that relies on “if x C z and y P z then x C y”, loosely “anything which causes the whole, causes the part”. Since there appear to be counterexamples to that as a causal intuiton, it’s a good idea to try to eliminate the step, even for a constructed relation based on C.
Here is a new version, which uses an alternative relation C’. The relation is still constructed on “entities”, under the assumption it makes some sense to talk of one entity as a cause of another (it could probably be adapted to other relata of causation like “events” or “states of affairs”). The conclusion of the argument turns out to be a bit weaker than before, but in a rather interesting way.
A1. The collection of all entities is a set E, with a causal relation C and a partial order P, such that x P y if and only if x is a part of y.
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.
Definitions: We define a relation C’ such that x C’ y iff there is some z which is part of x, but not part of y, and which satisfies z C y.
Note: This gives a relation which automatically satisfies “if x C’ y and x P w then w C’ y”, loosely “anything which is caused by a part is caused by the whole”. Also, C’ is guaranteed to be irreflexive since it is impossible to have x C’ x (no z can be simultaneously part of x and not part of x), loosely “no entity is a cause of itself”. These are plausible conditions on the underlying relation C, in which case we have C’ = C, but the construction of C’ means we don’t need to state these conditions as extra premises. We will see below that something else interesting can happen if C does not meet these conditions.
We then define a further relation ⇐ such that x ⇐ y iff x = y, or there are finitely many entities x1, …, xn such that x1 = x, xn = y and xi C’ xi+1 for i=1.. n-1.
Note: This construction ensures that ⇐ is a pre-order on E.
Say that x is “causally dependent” on y iff x is distinct from y but x ⇐ y; say that y is “dependent” iff there is some x on which y is causally dependent, and “independent” otherwise (this is equivalent to saying that any z ⇐ y satisfies z = y). Say that y is “wholly independent” if and only if y is not part of any dependent entity (which implies that y itself is independent). Say that a subset S of E is a “chain” iff for any x, y in S we have x ⇐ y or y ⇐ x. Say that S is an “endless chain” iff for any x in S there is some y in S such that x is causally dependent on y. Say that x is a “proper part” of y iff x is distinct from y but x P y.
A3. For any endless chain S in E, there is some entity z in E such that every x in S is a proper part of z.
Informally, the “endless chain” describes an odd causal relationship between entities : either an infinite descending sequence of causes, or a circle of causes. A natural question to ask is how there could be such a chain, and this involves treating the chain itself as some sort of entity to be explained.
Notice that if there are no endless chains at all, then A3 is true vacuously. Also, A3 is true provided arbitrary collections of entities can be aggregated together to form an entity, but the argument doesn’t need to rely on such a strong assumption. Finally, A3 is true if there is a “maximum” entity w, one of which every other entity is a proper part; but again the argument doesn’t need to rely on such an assumption.
Lemma 1: For any chain S in E, there is some entity w in E such that w ⇐ y for every y in S.
Proof: Suppose S is 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 A3, there is some z in E of which every x in S is a part. Now consider any y in S. There is some x not equal to y in S with x ⇐ y, so there is an entity x2, with x C’ x2 ⇐ y. Further, as x P z we have z C’ x2 ⇐ y and hence z ⇐ y.
Lemma 2: For any x in E, unless some w ⇐ x is wholly independent, then there is some dependent y ⇐ x, such that for any dependent z ⇐ y we have y ⇐ z.
Proof: This follows from Zorn’s Lemma for pre-orders. Consider the set Y = {y: y is dependent, and y ⇐ x}, and observe that for any chain S in Y, Lemma 1 gives some w with w ⇐ s for every s in S and also w ⇐ x. Either this w is wholly independent or it is part of some dependent y with y ⇐ s for every s in S and y ⇐ x, so that this y is also in Y. Zorn’s Lemma now implies that Y contains some element which is minimal with respect to ⇐ within Y. To conclude the proof, consider any other dependent z ⇐ y; then z ⇐ x as well, so that z is in Y and y ⇐ z.
Theorem 3: For any x in E, there is some w ⇐ x which is wholly independent.
Proof: Suppose that no w ⇐ x is wholly independent, take a y satisfying Lemma 2, and consider the set S = {s is dependent: s ⇐ y}. Since y is dependent, there is some w C’ y, and some dependent v with w P v, so that v C’ y, and hence v is not equal to y. Hence S contains at least two members, and by Lemma 2, y ⇐ s for every member of S, and so S is an endless chain. So by A3, and by assumption that no z ⇐ x is wholly independent, there is some dependent 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 some y ⇐ x is wholly independent.
Now for the “uniqueness” part. As before, we partition E into three subsets. I are the “inert” entities, which do not causally-depend on anything and have no causal dependencies themselves. (These could be abstract entities like numbers, propositions and so on). Formally I = {x in E: there is no y with x C’ y or y C’ x}. U are the “uncaused causes”—formally U = {x in E: there is no y with y C’ x, but there is z with x C’ z}. O are all the “other”, caused entities, so that formally O = {x in E: there is some y with y C’ x}. We will also let W be the “wholly uncaused causes”—formally W = {u in U: there is no x, y with u P y and x C’ y}.
B1. If S is any subset of W such that for any x, y in S we have x P y or y P x, (call such an S a “chain of parts”), then there is some entity z of which all members of S are parts.
B2. Suppose that y ⇐ x and z ⇐ x. Then there is some entity w ⇐ x such that: w ⇐ y or y P w; w ⇐ z or z P w.
Informally, the idea is that y and z can’t independently cause x without any further causal explanation. So there must be some common cause, however each of them may be part of that common cause.
Definition: Say that entities x and y are “causally-connected” if and only if x=y or there are finitely-many entities x=x1,..,xn=y with each xi C’ xi+1 or xi+1 C’ xi for i=1..n-1.
B3. Any two entities x, y in O are causally-connected.
Informally, O doesn’t “come apart” into disconnected components, such as a bunch of isolated universes. Premises B1-B3 turn out to be necessary for Theorem 6 to hold, as well as sufficient (see below). So they can’t be made any weaker!
Theorem 4: For any x in O, there is a unique entity f(x) in W such that: f(x) ⇐ x, and any y in U with y ⇐ x satisfies y P f(x).
Proof: For any x in O, define a subset W(x) = {y in W: y ⇐ x}; this is non-empty by Theorem 3. Consider any non-empty chain of parts S that is a subset of W(x). Since W(x) is a subset of W, then by B1, there is some z in E of which all members of S are parts, and such a z must also be in W. (If z is part of some dependent x then so is some s in S part of x, which is impossible because s is in W). Also since y ⇐ x for some member y of S, where y cannot be equal to x, and y P z, we have y C’ x2 ⇐ x for some x2, so z ⇐ x, as in the proof of Lemma 1, and z is also a member of W(x). Even if S is empty, we can take some z in W(x), and then trivially all members of S are parts of z. By application of Zorn’s Lemma to W(x), there is an element f(x) in W(x) which is maximal with respect to P i.e. there is no other y in W(x) with f(x) P y. Now, by B2, for any other y ⇐ x in U there must be some z ⇐ x with f(x) P z and y P z; this implies z is in W, and given f(x) is maximal in W(x) we have z = f(x) and so y P f(x). This makes f(x) the unique maximal element of W(x).
Theorem 5: For any x, y in O, f(x) = f(y) if and only if x and y are causally-connected.
Proof: It is clear that if f(x) = f(y) then x is causally-connected to y (just build a path backwards from x to f(x) and then forward again to y). Conversely, consider any two x, y in O:
a) If x C’ y, then f(x) is in U and satisfies f(x) ⇐ y so we have f(x) P f(y). Since x is not f(x), we have f(x) C’ x2 ⇐ x for some x2, and hence f(y) ⇐ x which means f(y) P f(x), and so f(x) = f(y).
b) If z is in U with z C’ x and z C’ y, then z P f(x) so f(x) ⇐ y and f(x) P f(y); similarly, f(y) P f(x) so that f(x) = f(y).
The result now follows by induction on the length of the causal path connecting x to y.
Theorem 6: If O is non-empty, then there is a single entity g in W such that: f(x) = g for every x in O, and y P g for every y in U.
Proof: Assuming O is non-empty, take any element y in O, and set g = f(y); then the result that f(x) = g for any x in O follows from Theorem 5 and B3; further, for any y in U, there is some x in O with y C’ x, so by Theorem 4, y P f(x). If there are no elements of O (meaning there are none in U either) then the Theorem is trivial.
Note that B1, B2 and B3 are entailed by the statement of Theorem 6. For B1, we can just take g as the relevant z. For B2, we can take g as the relevant w. B3 follows using using the first part of Theorem 5 (just track from x back to g, then forward to y again). The premises have been weakened a bit, but are still basically unjustifiable (we can imagine them being false without absurdity) and the above re-work shows they are needed for the uniqueness conclusion (Theorem 6).
One interesting fact is that Theorem 6 now has a loophole that wasn’t there before. If C is not equal to C’, then it is possible that there is some x C g, provided any such x is part of g (i.e. g contains any cause for itself within itself). This even allows Theorem 6 to be consistent with a premise like this:
B4: For any entity x, there is some y with y C x.
Informally, “every entity has a cause”, a premise which is usually considered a fatal inconsistency in a first cause argument! With a bit of renaming, the set O could be said to consist of “ordinary” entities (ones which have causes that are not parts of themselves) and the remaining non-inert entities are “extraordinary” entities (ones which contain all their own causes). W are “wholly extraordinary entities”, ones which are not part of any ordinary entity. Theorem 6 implies that every ordinary entity is causally dependent on a single wholly extraordinary entity, one which contains every other extraordinary entity.
Again, there aren’t any particularly theistic conclusions here, since g could very well be a maximum entity if there is one—say g is the whole universe or multiverse. In that case, g contains every ordinary entity as well as containing all the extraordinary entities.
I wonder if there should be some statute of limitations on that; whether, say, ten levels of positive-karma posts can protect against a higher-level negative-karma post?)
At the very least that many levels of positive votes in a row will likely insulate you from having the attempt to circumvent the troll tax summarily downvoted then banned. Since doing this isn’t too much additional work (and deep nesting is a nuisance in other ways anyhow) it probably suffices to culturally support or accept transplanted conversations that turn out to be valuable despite an early outlier. (It isn’t something that happens too often.)
Set Theory and Uncaused Causes
I’m relocating part of a thread that was originally on “Welcome to Less Wrong” but has wandered way off topic. It also seems that a remote ancestor comment was heavily downvoted, discouraging further contributions in the original place. So I’m moving into the Open thread.
Here are links to my latest version of the “recipe”, and to CCC’s response
As discussed, I have a new version which preserves the proof structure, but weakens the premises about as much as possible.
A1. The collection of all entities is a set E, with a causal relation C and a partial order P, such that x P y if and only if x is a part of y.
Note: This merges the assumption that P is a partial order into the overall set-up; that feature of P now gets used earlier in the argument.
A2. The set E can be well-ordered.
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.
Definitions: We define a relation C such that x C y iff there are entities v, w such that v P x, y P w and v C w.
Note: This gives a broader causal relation which automatically satisfies “if x C y and x P z then z C y” as well as “if x C z and y P z then x C y”, loosely “anything which is caused by a part is caused by the whole” and “anything which causes the whole, causes the part”. So we don’t need to state those as extra premises.
We then define a further relation ⇐ such that x ⇐ y iff x = y, or there are finitely many entities x1, …, xn such that x1 = x, xn = y and xi C* xi+1 for i=1.. n-1.
Note: This construction ensures that ⇐ is a pre-order on E.
Say that a subset S of E is a “chain” iff for any x, y in S we have x ⇐ y or y ⇐ x. Say that S is an “endless chain” iff for any x in S there is some y in S distinct from x with y ⇐ x. We shall say that y is “uncaused” if and only if there is no z in E distinct from y with z C* y (this of course implies there is no z distinct from y with z C y, but it also implies that y isn’t part of anything which is caused by something distinct from y). Say that x is a proper part of y iff x is distinct from y but x P y.
A3. 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 entity x in E such that x ⇐ y for every y in S.
Proof: Suppose S is 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 A3, there is some z in E of which every x in S is a part. Now consider any y in S. There is some x not equal to y in S with x ⇐ y, so there are entities x = x1… xn = y with each xi C xi+1 for i=1..n-1. Further, as x P z 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 every z ⇐ y, we must have y ⇐ z.
Proof: This follows from 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 A3 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.
We now partition E into three subsets. I are the “inert” entities, which do not cause anything and have no causes themselves. (Note that the new version allows there to be some of these, unlike the previous version; you can think of them as abstract entities like numbers, sets, propositions and so on, if you want to). Formally I = {x in E: there is no y distinct from x with x C y or y C x}. U are the “uncaused causes”—formally U = {x in E: there is no y distinct from x with y C x, but there is z distinct from x with x C z}. O are all the “other”, caused entities, so that formally O = {x in E: there is some y distinct from x with y C* x}.
B1. If S is any subset of U such that for any x, y in S we have x P y or y P x, (call such an S a “chain of parts”), then there is some entity z of which all members of S are parts.
B2. Suppose that y ⇐ x and z ⇐ x. Then there is some entity w such that: w ⇐ x; w ⇐ y or y P w; w ⇐ z or z P w.
(EDIT: Restated to ensure that Theorem 4 properly follows.) Informally, the idea is that y and z can’t independently cause x without any further causal explanation. So there must be some common cause, however each of them may be part of that common cause.
Definition: Say that entities x and y are causally-connected if and only if x=y or there are finitely-many entities x=x1,..,xn=y with each xi C xi+1 or xi+1 C xi for i=1..n-1.
B3. Any two entities x, y in O are causally-connected.
Informally, O doesn’t “come apart” into disconnected components, such as a bunch of isolated universes. Premises B1-B3 turn out to be necessary for Theorem 6 to hold, as well as sufficient (see below). So they can’t be made any weaker!
Theorem 4: For any x in O, there is a unique entity f(x) in U such that: f(x) ⇐ x, and any other y in U with y ⇐ x satisfies y P f(x).
Proof: For any x in O, define a subset U(x) = {y in U: y ⇐ x}; this is non-empty by Theorem 3. Consider any chain of parts S that is a subset of U(x). If it has at least two members, then by B1 there is some z in E of which all members of S are parts, and such a z must be in U. (If not, then note any w C z would also satisfy w C s for each member s of S, which would require them all to be equal to w). Also since y ⇐ x for any member of S and y P z we have z ⇐ x. So z is also a member of U(x). Or if S is a singleton—say {z} - then clearly all members of S are parts of z, and z is also in U(x). By application of Zorn’s Lemma to U(x), there is a P-maximal element f(x) in U(x) such that there is no other y in U(x) with f(x) P y. By B2, for any other y in U(x) there must be some z in U(x) with f(x) P z and y P z; given f(x) is maximal we have z = f(x) and so y P f(x). This makes f(x) the unique maximal element of U(x).
Theorem 5: For any x, y in O, f(x) = f(y) if and only if x and y are causally-connected.
Proof: It is clear that if f(x) = f(y) then x is causally-connected to y (just build a path backwards from x to f(x) and then forward again to y). Conversely, consider any two x, y in O:
a) If x C y, then f(x) is in U and satisfies f(x) ⇐ y so we have f(x) P f(y). Since x is not f(x), we have f(x) C x2 ⇐ x for some x2, and hence f(y) C* x2 ⇐ x i.e. f(y) ⇐ x which means f(y) P f(x), and so f(x) = f(y).
b) If z is in U with z C x and z C y, then z P f(x) so f(x) ⇐ y and f(x) P f(y); similarly, f(y) P f(x) so that f(x) = f(y).
The result now follows by induction on the length of the causal path connecting x to y.
Theorem 6: If O is non-empty, then there is a single entity g in U such that: f(x) = g for every x in O, and y P g for every y in U.
Proof: Assuming O is non-empty, take any element y in O, and set g = f(y); then the result that f(x) = g for any x in O follows from Theorem 5 and B3; further, for any y in U, there is some x in O with y C* x, so by Theorem 4, y P f(x). If there are no elements of O (meaning there are none in U either) then the Theorem is trivial.
Finally, note that B1, B2 and B3 are entailed by the statement of Theorem 6. For B1, we can just take g as the relevant z. For B2, we can take g as the relevant w. B3 follows using using the first part of Theorem 5 (just track from x back to g, then forward to y again).
I’m just about done now, so unless there are errors in the above proof will leave it. What are the residual weak points? Well, B2 and B3 have been weakened a bit, but are still basically unjustifiable (we can imagine them being false without absurdity) and the above re-work shows they are needed for the uniqueness conclusion (Theorem 6). Also, we have the weakness of not deriving anything else useful about g.
This will lead to a problem.
Consider assumption A3:
Consider any endless chain consisting of at minimum two elements. Consider two elements in that chain, x and y, such that x C y. x and y are both proper parts of z. Therefore, x C z, and z C y. But then we have a longer chain; using x C z C y in place of x C y. Each element of that longer chain must then, by A3, be a proper part of a larger entity, z2. But then, similarly, we can construct the chain using x C z C z2 C* y. There are therefore an infinite number of entities z, z2, z3, z4… and so on, each including the one before it as a proper part (and nothing that is not part of the one before it).
Furthermore, anything which is a proper part of anything else is a part of such an infinitely recursive loop by default.
This leads to trouble in the proof of theorem 3.
It follows that z C y but it does not follow that x C z or that y C z.. The “whole” z may be a cause of its parts, without in turn being caused by its parts. Note that by construction of C it is true that if x is a cause of y and x is a part of z, then z C y. However, it is not generally true that if x is a cause of y and z is a part of x then z C y.
As an example of the intuition behind this: suppose I have a thermostat box containing two circuit boards. Board 1 is connected into my home heating system; Board 2 is a spare not connected into anything. It is true that Board 1 causes my heating to come on. It is true that the thermostat (of which Board 1 is part) causes my heating to come on. It is false that Board 2 (which is part of the thermostat) causes my heating to come on.
You are right that when adding z, we now get a longer chain {x, y, z}, but this won’t in general be an “endless chain” (the new z may well be an end).
It does, because y P z.
x C y, and y P z. Therefore, x C z.
No, you need x C y and z P y to get x C z (be careful about which way the P relation is going).
The intuition is “Anything which is a cause of the whole is a cause of the part”, not “Anything which is a cause of the part is a cause of the whole”. Again, there are intuitive examples here. (Compare me baking a cake for a child’s birthday party vs just buying the cake from a shop, and putting a few sprinkles and candles on the top. In the second case, I am a cause of some part of the cake as presented to the child, but not the whole cake, and if someone says “Wow that cake tasted delicious!” I’d have to admit I didn’t make it, only decorated it).
I make a cake. I am a cause of the cake. The cake contains eggs. I am not the cause of the eggs. I think “what causes the whole, causes each part” is a bad intuition to have.
In general, I think it is an error, and a source of confusion, to think of things rather than events having, or being, causes. I know people sometimes do, and I’ve gone along with it in #1 above, but I think it’s a mistake.
Why would anyone assume A3? It seems really arbitrary. Exception: you might believe A3 because you believe in an entity of which all others are parts. See below.
If E includes an entity V of which all others are parts (call it “the universe”) then, provided C is reflexive, V C* anything-you-like. And I think it’ll then turn out that the way all your theorems work is that V is the canonical uncaused cause of everything. Which is a bit dull and wouldn’t satisfy many theists. Perhaps something more interesting happens if you make C irreflexive instead, so that things don’t count as causes of themselves.
Fair points, though there is in fact a lot of disagreement about what are the basic relata of the causal relation: see the SEP entry for example. When we apply causation to entities (which we can sometimes do, as in your example) then “A causes B” probably means something like “at least one event in which A is involved is a cause of every event in which B is involved”.
On counterexamples to “what causes the whole, causes the part” : possibly an even stronger counterexample considers just one of the atoms in the cake. However, we must be careful here: it is only some temporal part of the egg (or of the atom) which is part of the cake; the eggs/atoms in their full temporal entirety are NOT parts of the cake in its full temporal entirety. We could perhaps treat the relevant temporal part (“egg mixed into cake” or “atom within cake”) as an “entity” in its own right, but then it does seem that by making the cake, I am a cause of all the events which involve that particular “entity” (since I put the egg/atom into the cake in the first place).
In any case, note that the most recent version of the argument doesn’t actually need to assume this “cause-whole ⇒ cause-part” applies to C, since it only ever uses the constructed relation C instead. The conclusion is still interesting, since if nothing Cs the entity g, then nothing Cs it either, and if g causes some whole of which each entity is a part, that is still an interesting property of g. The argument makes no assumptions on whether C is reflexive or not.
On A3, I’m not totally sure of the circumstances in which we can aggregate entities together and treat them as parts of a single entity, but if the entities are causally related (and particularly if they are causally-related in an odd way, like an endless chain), then it does make some sort of sense to do this aggregation. After all, we immediately want to ask the question “How could there be an endless chain?” a question which does treat the “chain” as some sort of an entity to be explained. If entities are not causally related (they are in different universes), lumping them together seems much less natural.
Finally, on the “maximal entity” approach, CCC I believe discussed this in the original thread before I lifted here, and he seems to find it theologically interesting.
I agree that “x C y, and y P z. Therefore, x C z” is wildly unintuitive, causes problems, and is just plainly wrong. But...
...
...actually, looking back, you’re right. I apologise; I misread the definition of C* (I read w P y instead of y P w).
I’m going to have to look through it again before I can comment further.
Looking over the “recipe” again, I notice one thing I failed to notice earlier; at no point have you formally defined the uncaused cause as God. That’s a weak step, but if you actually want to use that argument to argue for monotheism (instead of merely the presence of an uncaused entity, nature unknown) then I think it’s a necessary step. Some of the assumptions necessary to eliminate multiple uncaused causes are a bit weak, but I think that you have a very good argument that somewhere in the history of the universe there must be at least one uncaused cause of some sort.
No, I omitted that step for reasons discussed in the earlier thread: this gives too weak a “God” to be any interest to anyone, and is downright confusing.
The only way I can think to get back to some form of traditional theism is to add a premise saying that “every entity not of type G has a cause” (insert your favourite G) and then perhaps to pull the modal trick of claiming all the premises are possible...
Okay, that’s reasonable.
I have spotted an error in the statement (and proof) of Theorem 5, and then Corollary 6. The issue is that for any uncaused y we must have f(y) = y, so if there are several uncaused entities then they can’t all have f(y) = g. The revised statements should go like this:
Theorem 5: Let x and y both have causes. Then f(x) = f(y) if and only if x and y are causally connected.
Proof: It is clear that if f(x) = f(y) then x is causally-connected to y (just build a path backwards from x to f(x) and then forward again to y). Conversely, suppose that x C y, then f(x) is uncaused and satisfies f(x) ⇐ y so we have f(x) P f(y); since x is caused, there are f(x)=x1,...,xn=x such that each xi C xi+1 for i=1..n-1, then by A3 we have f(y) C x2 and hence f(y) ⇐ x, which implies f(y) P f(x) and so by B1 f(x) = f(y). Next, suppose that for some uncaused z we have z C x and z C y; then z P f(x) which implies by A3 that f(x) C y and hence f(x) P f(y); similarly, f(y) P f(x) so by B1 f(x) = f(y). By an induction on the length of any other path connecting x to y, we have that f(x) = f(y).
Corollary 6: There is a single g in E such that: f(x) = g for every x in E with a cause, and every uncaused y P g.
Proof: Suppose there is at least one entity x with a cause, then set g = f(x). For any other caused entity y, f(y) = g by Theorem 5 and B5, and for an uncaused y, B5 implies y ⇐ x, so that y P g. Lastly, if there are no caused entities, then B5 implies that E = {y} for some uncaused y, so we can just pick g = y.
I have also spotted a way of weakening or removing some of the premises (in particular A3, and B1 to B4). I will update with that later today.
I’ve had another look at the argument, and spotted a way to remove the step that relies on “if x C z and y P z then x C y”, loosely “anything which causes the whole, causes the part”. Since there appear to be counterexamples to that as a causal intuiton, it’s a good idea to try to eliminate the step, even for a constructed relation based on C.
Here is a new version, which uses an alternative relation C’. The relation is still constructed on “entities”, under the assumption it makes some sense to talk of one entity as a cause of another (it could probably be adapted to other relata of causation like “events” or “states of affairs”). The conclusion of the argument turns out to be a bit weaker than before, but in a rather interesting way.
A1. The collection of all entities is a set E, with a causal relation C and a partial order P, such that 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.
Definitions: We define a relation C’ such that x C’ y iff there is some z which is part of x, but not part of y, and which satisfies z C y.
Note: This gives a relation which automatically satisfies “if x C’ y and x P w then w C’ y”, loosely “anything which is caused by a part is caused by the whole”. Also, C’ is guaranteed to be irreflexive since it is impossible to have x C’ x (no z can be simultaneously part of x and not part of x), loosely “no entity is a cause of itself”. These are plausible conditions on the underlying relation C, in which case we have C’ = C, but the construction of C’ means we don’t need to state these conditions as extra premises. We will see below that something else interesting can happen if C does not meet these conditions.
We then define a further relation ⇐ such that x ⇐ y iff x = y, or there are finitely many entities x1, …, xn such that x1 = x, xn = y and xi C’ xi+1 for i=1.. n-1.
Note: This construction ensures that ⇐ is a pre-order on E.
Say that x is “causally dependent” on y iff x is distinct from y but x ⇐ y; say that y is “dependent” iff there is some x on which y is causally dependent, and “independent” otherwise (this is equivalent to saying that any z ⇐ y satisfies z = y). Say that y is “wholly independent” if and only if y is not part of any dependent entity (which implies that y itself is independent). Say that a subset S of E is a “chain” iff for any x, y in S we have x ⇐ y or y ⇐ x. Say that S is an “endless chain” iff for any x in S there is some y in S such that x is causally dependent on y. Say that x is a “proper part” of y iff x is distinct from y but x P y.
A3. For any endless chain S in E, there is some entity z in E such that every x in S is a proper part of z.
Informally, the “endless chain” describes an odd causal relationship between entities : either an infinite descending sequence of causes, or a circle of causes. A natural question to ask is how there could be such a chain, and this involves treating the chain itself as some sort of entity to be explained.
Notice that if there are no endless chains at all, then A3 is true vacuously. Also, A3 is true provided arbitrary collections of entities can be aggregated together to form an entity, but the argument doesn’t need to rely on such a strong assumption. Finally, A3 is true if there is a “maximum” entity w, one of which every other entity is a proper part; but again the argument doesn’t need to rely on such an assumption.
Lemma 1: For any chain S in E, there is some entity w in E such that w ⇐ y for every y in S.
Proof: Suppose S is 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 A3, there is some z in E of which every x in S is a part. Now consider any y in S. There is some x not equal to y in S with x ⇐ y, so there is an entity x2, with x C’ x2 ⇐ y. Further, as x P z we have z C’ x2 ⇐ y and hence z ⇐ y.
Lemma 2: For any x in E, unless some w ⇐ x is wholly independent, then there is some dependent y ⇐ x, such that for any dependent z ⇐ y we have y ⇐ z.
Proof: This follows from Zorn’s Lemma for pre-orders. Consider the set Y = {y: y is dependent, and y ⇐ x}, and observe that for any chain S in Y, Lemma 1 gives some w with w ⇐ s for every s in S and also w ⇐ x. Either this w is wholly independent or it is part of some dependent y with y ⇐ s for every s in S and y ⇐ x, so that this y is also in Y. Zorn’s Lemma now implies that Y contains some element which is minimal with respect to ⇐ within Y. To conclude the proof, consider any other dependent z ⇐ y; then z ⇐ x as well, so that z is in Y and y ⇐ z.
Theorem 3: For any x in E, there is some w ⇐ x which is wholly independent.
Proof: Suppose that no w ⇐ x is wholly independent, take a y satisfying Lemma 2, and consider the set S = {s is dependent: s ⇐ y}. Since y is dependent, there is some w C’ y, and some dependent v with w P v, so that v C’ y, and hence v is not equal to y. Hence S contains at least two members, and by Lemma 2, y ⇐ s for every member of S, and so S is an endless chain. So by A3, and by assumption that no z ⇐ x is wholly independent, there is some dependent 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 some y ⇐ x is wholly independent.
Now for the “uniqueness” part. As before, we partition E into three subsets. I are the “inert” entities, which do not causally-depend on anything and have no causal dependencies themselves. (These could be abstract entities like numbers, propositions and so on). Formally I = {x in E: there is no y with x C’ y or y C’ x}. U are the “uncaused causes”—formally U = {x in E: there is no y with y C’ x, but there is z with x C’ z}. O are all the “other”, caused entities, so that formally O = {x in E: there is some y with y C’ x}. We will also let W be the “wholly uncaused causes”—formally W = {u in U: there is no x, y with u P y and x C’ y}.
B1. If S is any subset of W such that for any x, y in S we have x P y or y P x, (call such an S a “chain of parts”), then there is some entity z of which all members of S are parts.
B2. Suppose that y ⇐ x and z ⇐ x. Then there is some entity w ⇐ x such that: w ⇐ y or y P w; w ⇐ z or z P w.
Informally, the idea is that y and z can’t independently cause x without any further causal explanation. So there must be some common cause, however each of them may be part of that common cause.
Definition: Say that entities x and y are “causally-connected” if and only if x=y or there are finitely-many entities x=x1,..,xn=y with each xi C’ xi+1 or xi+1 C’ xi for i=1..n-1.
B3. Any two entities x, y in O are causally-connected.
Informally, O doesn’t “come apart” into disconnected components, such as a bunch of isolated universes. Premises B1-B3 turn out to be necessary for Theorem 6 to hold, as well as sufficient (see below). So they can’t be made any weaker!
Theorem 4: For any x in O, there is a unique entity f(x) in W such that: f(x) ⇐ x, and any y in U with y ⇐ x satisfies y P f(x).
Proof: For any x in O, define a subset W(x) = {y in W: y ⇐ x}; this is non-empty by Theorem 3. Consider any non-empty chain of parts S that is a subset of W(x). Since W(x) is a subset of W, then by B1, there is some z in E of which all members of S are parts, and such a z must also be in W. (If z is part of some dependent x then so is some s in S part of x, which is impossible because s is in W). Also since y ⇐ x for some member y of S, where y cannot be equal to x, and y P z, we have y C’ x2 ⇐ x for some x2, so z ⇐ x, as in the proof of Lemma 1, and z is also a member of W(x). Even if S is empty, we can take some z in W(x), and then trivially all members of S are parts of z. By application of Zorn’s Lemma to W(x), there is an element f(x) in W(x) which is maximal with respect to P i.e. there is no other y in W(x) with f(x) P y. Now, by B2, for any other y ⇐ x in U there must be some z ⇐ x with f(x) P z and y P z; this implies z is in W, and given f(x) is maximal in W(x) we have z = f(x) and so y P f(x). This makes f(x) the unique maximal element of W(x).
Theorem 5: For any x, y in O, f(x) = f(y) if and only if x and y are causally-connected.
Proof: It is clear that if f(x) = f(y) then x is causally-connected to y (just build a path backwards from x to f(x) and then forward again to y). Conversely, consider any two x, y in O:
a) If x C’ y, then f(x) is in U and satisfies f(x) ⇐ y so we have f(x) P f(y). Since x is not f(x), we have f(x) C’ x2 ⇐ x for some x2, and hence f(y) ⇐ x which means f(y) P f(x), and so f(x) = f(y).
b) If z is in U with z C’ x and z C’ y, then z P f(x) so f(x) ⇐ y and f(x) P f(y); similarly, f(y) P f(x) so that f(x) = f(y).
The result now follows by induction on the length of the causal path connecting x to y.
Theorem 6: If O is non-empty, then there is a single entity g in W such that: f(x) = g for every x in O, and y P g for every y in U.
Proof: Assuming O is non-empty, take any element y in O, and set g = f(y); then the result that f(x) = g for any x in O follows from Theorem 5 and B3; further, for any y in U, there is some x in O with y C’ x, so by Theorem 4, y P f(x). If there are no elements of O (meaning there are none in U either) then the Theorem is trivial.
Note that B1, B2 and B3 are entailed by the statement of Theorem 6. For B1, we can just take g as the relevant z. For B2, we can take g as the relevant w. B3 follows using using the first part of Theorem 5 (just track from x back to g, then forward to y again). The premises have been weakened a bit, but are still basically unjustifiable (we can imagine them being false without absurdity) and the above re-work shows they are needed for the uniqueness conclusion (Theorem 6).
One interesting fact is that Theorem 6 now has a loophole that wasn’t there before. If C is not equal to C’, then it is possible that there is some x C g, provided any such x is part of g (i.e. g contains any cause for itself within itself). This even allows Theorem 6 to be consistent with a premise like this:
B4: For any entity x, there is some y with y C x.
Informally, “every entity has a cause”, a premise which is usually considered a fatal inconsistency in a first cause argument! With a bit of renaming, the set O could be said to consist of “ordinary” entities (ones which have causes that are not parts of themselves) and the remaining non-inert entities are “extraordinary” entities (ones which contain all their own causes). W are “wholly extraordinary entities”, ones which are not part of any ordinary entity. Theorem 6 implies that every ordinary entity is causally dependent on a single wholly extraordinary entity, one which contains every other extraordinary entity.
Again, there aren’t any particularly theistic conclusions here, since g could very well be a maximum entity if there is one—say g is the whole universe or multiverse. In that case, g contains every ordinary entity as well as containing all the extraordinary entities.
At the very least that many levels of positive votes in a row will likely insulate you from having the attempt to circumvent the troll tax summarily downvoted then banned. Since doing this isn’t too much additional work (and deep nesting is a nuisance in other ways anyhow) it probably suffices to culturally support or accept transplanted conversations that turn out to be valuable despite an early outlier. (It isn’t something that happens too often.)