More About Sets

There are of course lots of doubletons, such as {1,2} and {13,666}, and so on for other cardinalities. For our purposes however we are not interested what particular elements each Chu space contains but only their number and their interactions if any. Sets, or discrete spaces, are those spaces whose points do not interact at all but are free to "move" entirely independently. All sets of the same cardinality are isomorphic, and since isomorphism is all we care about we shall name sets according to their cardinality.

"Discrete" and "set" are synonymous as descriptions of Chu spaces: they denote a Chu space having all possible columns, and connote the complete absence of structure. A set with n points is the minimally constrained n-point space, having all 2n states, and it is reasonable to identify it with the number n. Nondiscrete Chu spaces have fewer than 2n states.

Intrinsically, states can be understood equally well as n-tuples over 2 or as functions from the set n = {0,1,...,n-1} to the set 2. Extrinsically, states are possibilities: they can be viewed as the possible states or "paintings" of the space using colors from the palette 2, or as the possible predicates on it, or as its possible subspaces. In the case of a set, its subspaces are simply its subsets, of which there are 2n.

The state interpretation of columns gives Chu spaces some of the qualities of a state automaton, but there are several important differences.

First, these automata can branch and exhibit concurrent behavior but cannot loop other than by being infinite. In this respect they are like formal languages, but with a more local notion of choice (one only chooses a whole string from a language before uttering it, whereas branching permits such choices to be made in mid-stream). Unlike formal languages

Second, there are no explicitly given transitions. Instead the inclusion relation between states determines which states are accessible from which. The state 101 is accessible from 001 but not from 011 because the middle event of the latter is 1 meaning that event has happened, and it would not make sense to pass to a state which believed that it had not happened.

Third, there is no alphabet; Chu spaces correspond to unlabeled event structures, whose events are anonymous. They can be converted to labeled event structures by equipping them with a labeling function from events to actions, but that's another story.

Sets are the ultimate in concurrent processes: their events are completely independent and could be parallel events within a single computer chip or spread out over the galaxy, with no interaction in either case.