Peter Freyd wrote:
Vaughn's point is a good one, but not quite apropos of Tom's notion of self-similarity. In the category of ordered sets the final co-algebra of the Pavlovic-Pratt functor is, indeed, the half-open real interval, [0,1). But Tom was in the category of topological spaces. To make [0,1) into a self-similar _space_ one must reinterpret their functor so that the new point of the one-point compactification of each copy of X is identified with the bottom of the next copy.
the title of tom's series of papers suggests that he wants to develop a General Theory of Self-Similarity, not just a topological theory. and in a general theory, the various decompositions of [0,1) into the copies of [0,1) and so on --- would qualify as self-similarity in most people's eyes. i didn't read tom's papers yet, but it seems to me that the essence of self-similarity in general, and the conceptual value of all of our reconstructions of the reals using self-similarity, is not so much in their topology or order, but in their coalgebra, capturing the infinite subdivisions. and finally, to pop up a level, i am a bit surprised with tom's line of reasoning: "it's not topology, so it shouldn't be mentioned", tacitly supported by peter's posting. i was under the impression that the authors in all sciences consider it to be a matter of good taste to be generous in references and acknowledgements, and not frugal; to be inclusive and not exclusive. people provide as many references as they can, to let the readers track their ideas, and make the connections. useful papers don't cite just their parent papers, but also their grandparent papers, and brothers and sisters. especially when the parents so clearly and generously recognize their debts, as peter's original postings did. -- dusko 25-Nov-2004 14:18:16 -0400,1422;000000000000-00000000