Dusko Pavlovic <dusko@kestrel.edu> wrote:
To begin embarassing myself --- I am not sure what Peano's singleton operation is. Is it the map x|-->{x}?
If it is --- why is this operation totally arbitrary, like Bill says? In particular, why is it more arbitrary than the successor operation in arithmetic (which Peano used)? It comes about as a part of the initial algebra structure on Godel's cumulative hierarchy, just like the successor comes about in NNO.
Well, something *like* Peano's operation occurs in that initial algebra structure, but that is not much to the point. The point is: successor did not have to wait for NNOs to be defined. It occurs throughout arithmetic for nearly as long as we have records of systematic thought. And it is central to all uses of arithmetic today. The *singleton subset* idea is also very old: A geometric condition can define a subset of points in the plane, and perhaps a singleton subset. And singleton subsets are all over math today for the same reason. It is a recent idea that given any set x there is some set {x}. Bill traces it to Peano. It plays no role in ordinary mathematical practice, and is unnecessary in set theory. It does not exist in categorical set theory.
Also, I somehow came to think of set theory as *tree representations of abstract sets*, much like vector spaces are used for group representations. Is this wrong? It seems to me that introducing the external, "spurious" elements (eg vectors) is the whole point of representations.
The whole point of group representations is that each group has many of them. The classical Lie groups are given as groups of linear transformations in the first place. The power of representation theory is to relate these with *other* representations of the same groups. Each ZF set has exactly one membership tree. Thus the "representation" cannot do anything like what group representations do. And obviously it plays no role in ordinary math practice. I hope no one believes that singletons, or trees, or vectors are "spurious" per se. Some uses of the ideas are "arbitrary", and some claims about them are "spurious". best, Colin