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.
Informally, "a cause of the whole is a cause of any part".
B4. Suppose that y <= x and z <= x and both y, z are uncaused. Then y P z or z P y, or there is some w of which both y and z are proper parts.
Informally, two uncaused y and z can't independently conspire to cause x unless they are parts of a common entity.
Definition: Say that entities x and y are causally-connected if and only if x = y, or there are entities x=x1,..,xn=y with either xi C xi+1 or xi+1 C xi for each i=1..n-1.
B5. Any two entities in E are causally-connected.
Informally, E doesn't "come apart" into completely disconnected components, such as a bunch of isolated universes.
Theorem 4: For any x in E, there is a unique entity f(x) in E such that: f(x) is uncaused, f(x) <= x, and any other uncaused y with y <= x satisfies y P f(x).
Proof: For any x, define a subset E' = {y in E: y <= x, y is uncaused}. Consider any chain of parts S in E' with at least two elements. By B2 there is some z in E of which all members of S are parts. By B3, z must be uncaused (or else some w C z would also be a cause of all the members of S, which would require them all to be equal to w, so S would be a singleton), and by A3, z <= x. So z is also a member of E'. By application of Zorn's Lemma to E', there is a P-maximal element f in E' such that there is no other y in E' with f P y. But then, by B4, for any y in E' we must have y P f; this makes f unique.
Theorem 5: For any x, y in E, 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). This implies f(x) = f(y). By a simple induction on n we have that if x is causally-connected to y, then f(x) = f(y).
Corollary 6: There is a single entity g in E such that f(x) = g for every entity x in E.
Proof: This follows from Theorem 5 and B5.
Done!
(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?)
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.
An interesting assumption. Necessary for theorem 3, but I suspect that it'll mean that the original cause described in theorem 3 will then very probably be an en...
A few notes about the site mechanics
A few notes about the community
If English is not your first language, don't let that make you afraid to post or comment. You can get English help on Discussion- or Main-level posts by sending a PM to one of the following users (use the "send message" link on the upper right of their user page). Either put the text of the post in the PM, or just say that you'd like English help and you'll get a response with an email address.
* Normal_Anomaly
* Randaly
* shokwave
* Barry Cotter
A note for theists: you will find the Less Wrong community to be predominantly atheist, though not completely so, and most of us are genuinely respectful of religious people who keep the usual community norms. It's worth saying that we might think religion is off-topic in some places where you think it's on-topic, so be thoughtful about where and how you start explicitly talking about it; some of us are happy to talk about religion, some of us aren't interested. Bear in mind that many of us really, truly have given full consideration to theistic claims and found them to be false, so starting with the most common arguments is pretty likely just to annoy people. Anyhow, it's absolutely OK to mention that you're religious in your welcome post and to invite a discussion there.
A list of some posts that are pretty awesome
I recommend the major sequences to everybody, but I realize how daunting they look at first. So for purposes of immediate gratification, the following posts are particularly interesting/illuminating/provocative and don't require any previous reading:
More suggestions are welcome! Or just check out the top-rated posts from the history of Less Wrong. Most posts at +50 or more are well worth your time.
Welcome to Less Wrong, and we look forward to hearing from you throughout the site!
Note from orthonormal: MBlume and other contributors wrote the original version of this welcome post, and I've edited it a fair bit. If there's anything I should add or update on this post (especially broken links), please send me a private message—I may not notice a comment on the post. Finally, once this gets past 500 comments, anyone is welcome to copy and edit this intro to start the next welcome thread.