Saturday, July 07, 2007

An Empty Set?

My problem with singletons was (as previously mentioned) that although it is intuitively clear how a plurality (many things) could also be a unity (one plurality)—e.g. one deck of 52 cards—it is consequently obscure how a single object could similarly be a unity different from that object. Nonetheless there is a related obscurity with pluralities, e.g. how do two pairs of socks differ from those four socks?
......Although our 52 cards (think of any particular deck) are exactly the same as those 4 suits (hearts, clubs, diamonds, spades) of 13 cards each—an application of 4 times 13 being 52—those are two different pluralities (52 cards and 4 suits), two different numbers of (different sorts of) things. In one sense they are the same thing, the same deck, but in another sense they are two different things, two different ways of partitioning the cards. When we refer to a collection—e.g. to a par of socks—we are usually referring not to that partition but to those things—to those two socks—but still, two pairs of socks are one collection, while the same four socks are another.
......Perhaps that is why collections are regarded as abstract objects, because they are akin to partitions (which are clearly abstract objects). And when we compare partitions it makes sense to compare partitions of some given things, so perhaps we should be thinking of partitions of some totality of things that are not collections, some universe (of discourse). After all, it remains fairly mysterious what something being a thing (standing apart from everything else) amounts to, but presumably whatever it is will involve those other things (that other stuff) to some extent.
......Of course, such partitions are just collections of collections, so I’m presupposing collections (of many into one), but if collections are indeed akin to partitions then that might shed some light on singletons—e.g. were each collection all the parts but one of some such universal partition, with everything else (in the universe) in the remaining part, so that a singleton would be a part of a partition. Or perhaps the singleton would be the whole partition (an abstract object), containing its single member (that part of the partition) within the rest of itself (the set-theoretic lasso).
......Now, so far I’ve overlooked overlaps, e.g. the collection of pairs (two aces, two fours etc.) is not a partition of our 52 cards, but of three identical decks—or rather (that being impossible) I should not have excluded collections from the universe. Things would get pretty complicated were they all included to begin with, but something like the standard cumulative-hierarchical approach might work out; and if so then maybe the (non-formal) empty set would be the degenerate partition of (i.e. nothing but the lasso that is) some such cumulative-hierarchical universe?

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