User talk:Lulu of the Lotus-Eaters/odd creature

An odd creature

Some months ago I created the article List of every Wikipedia list that does not contain itself. This is in the spirit of part of the discussion in this article, the more precise in the self-reference. In any case, it simply redirects to Russell's paradox; it might be something like a slight easter egg on Wikipedia, but please leave it be (it may not be a brillant joke, but it's worth a couple tens of bytes in the WP database.

Updating my userpage, an odd little self-reference occurred to me: List of every Wikipedia list that contains more items than this list. I know this is a digression from article discussion, but I'm trying to get a handle on exactly what kind of creature this hypothetical article is. It's not quite a Russell paradox, nor quite a Curry paradox.

Here's the issue, in case it's not immediately obvious: whether or not this list is "a problem" depends on the world external to its definition. If the world is certain ways, it's a perfectly ordinary collection. If the world is other ways, it's a paradox. Let's demonstrate:

• Suppose Wikipedia contains ten lists: five of them list 3 items each; five of them list 20 items each. No problem at all arises here, we just put the five big lists on the new list (bringing Wikipedia to eleven lists total, one of them containing five items).

• On the other hand, suppose Wikipdedia contains ten list: four of them list 3 items each; six of them list 5 items each. Now we have a problem. If we include the 5-items lists, this list has 6 items, and all the 5-item lists must be removed. If we leave off the 5-item lists, this list has zero items, and all the 5-item lists must be added (we might initially add the 3-item lists as well, but taking them off once this grows creates no special problem).

Is there a well-known name for logical/empirical paradoxes of this sort? I.e. ones that are only contingently paradoxical? LotLE×talk 06:32, 20 June 2006 (UTC)

Possible worlds

Hmm... There are plenty of stable states that this list could take. ie. say thre were 4 lists of 3 items, and 10 lists of 15 items, we'd be fine. To me this seems rather ordinary, "paradoxically" speaking... consider the "list of points at which lines ax+b and cx+d intersect?" depending on the state of a b c and d, we could have many solutions, one solution, or no solution... which seems to be exactly the same sort of outcome as this list you've just proposed. A good mathemetician might be able to generalize it with a formula, or failing that there's always computer assisted exhaustion to fall back on. - Rainwarrior 13:04, 20 June 2006 (UTC)

Certainly, there are many states of the world, or "possible worlds" where the paradox does not arise. In fact, I'm pretty sure that in a measure theoretic sense, the measure of the set of possible worlds where the paradox does arise is zero. Nonetheless, one can easily find an enumerably infinite set of "problem" possible worlds. Let's call worlds that are paradoxical in the described sense "L-paradoxical". The second part of this is almost immediate, by a trivial variation of the prior example:

• Suppose Wikipedia contains ten lists: four of them list 3 items; five of them list 5 items; one of them, BIG, lists "many" items. For every value of N > 5, the possible world described is L-paradoxical. That is, BIG must surely be included in "LoeWltcmittl". But if we try to put all the 5-item lists in LoeWltcmittl, we get a problem. So there's a countable infinity of problems. (and infinitely many other families of "problem worlds" are easy to construct).

The measurement thing is slightly more involved, but not too much. Basically, a possible world (down to homomorphism) is defined by a set of ordered pairs of natural numbers: <NumMembers, SizeOf>. That is, a (homomorphic equivalence class of) world(s) is described by the number of lists of each size that are in it. For example: "4 3-item; 5 5-item; 1 20-item". So described, some worlds are L-paradoxical, and others are not. Notice that the possible worlds are enumerable.

• We can consider all the worlds ranked by total size: worlds of 1 list (or whatever size), worlds of 2 lists, etc. If a world of size M has an "instability point"—that is, a number of items in LoeWltcmittl that would create an L-paradox—that point must be some number ≤ M. If every list in the world contains more items than M, no instability is possible. However, since M is some particular finite number, the measure of M-sized worlds in which no list is as short as M is exactly 1 (natural numbers keep going up, after all, any initial segment is measure 0).

In practical terms, one might be surprised to find billion, or trillion, or googleplex length lists on Wikipedia, but formally there is no size bound. Of course, some practical system, like WP that has an extreme bias towards "small" lists (for any value of small; say, N < number of particles in the universe) is likely to be L-paradoxical with measure greater than zero.

But all of that is not really what I was asking. My original point was that it is interesting that a construct is not just empty, or just undefined in a simple way, in some possible worlds; rather the construct is paradoxical in some possible worlds... but perfectly ordinary in other possible worlds. In some worlds (infinitely many, in fact), LoeWltcmittl cannot contain any particular number of items (including zero), and yet it gives a precise inclusion criterion for any particular possible member.

I'm familiar with paradoxes that are paradoxical by their actual form, i.e. in every possible world. But it is somewhat novel to me to have stumbled on a paradox that is, as I say, contingently paradoxical. Of course, I'm sure someone has thought of this type of thing before... so I was just hoping to learn that this was already known as, e.g. "Jones' Paradox". LotLE×talk 19:14, 20 June 2006 (UTC)