Brian Clegg. A Brief History of Infinity (2003)

     Brian Clegg. A Brief History of Infinity: The Quest to Think the Unthinkable (2003) Well done, sometimes text-bookish, account of the history of the concept of infinity. Clegg is very good at potted biographies, and has a good grasp of the arc of developing understanding. He speculates perhaps a bit too much about the personalities and the tendency of thinkers about infinity to show signs of incipient or real madness.
     The notion of infinity has now, after the invention and development of set theory, a good logical foundation, but there are still conundrums worth pursuing. Clegg’s account of Russell’s paradox set me to thinking about the difference between sets and their elements. The questions is, does it make sense to speak of the type of a set or of its elements? If so, is the type of a set necessarily that of its elements? If not, then supersets need not be the same type as the sets that are its elements. There is perhaps a hint of this in the fact that the set of all subsets of a set is of larger size than the set itself. Anyhow, if a set and its elements are not of the same type, then Russell’s paradox dissolves. Or so it seems to me.
      More formally: define a simple set S(e) as one whose elements e are not themselves sets. Define the superset S’(S(e)) as the set whose members are S(e) and all its subsets. BTW, if S(e) is finite, then so is S’(S(e)). If S(e) is infinite, then S’(Se)) is its power set. We define the type of set as the type of its elements. Thus, a simple set is of the same type as its elements.
     The question I now ask is whether S and S’ are of the same type. I have defined the type of a set as the type of elements which are its members. Thus H(h) = “all human beings” by definition is type h, where h = “human being”. All its subsets will also be of type h. But what about its superset H’(H(h))? Is it of type h? IOW, is it true that H(h) –> H’(h)? It seems to me that this is not a necessary consequence. For while H(h) is of type H, H’ is of type “set”. IOW, I suspect (but cannot prove) that H’ is an axiomatic claim. It amounts to saying that a set may be subset of itself. Suppose we deny that. Then I think Russell’s paradox dissolves. Let S(-s) = “Sets that do not contain themselves.” Then if S’(S(-s)) does not imply S’(-s), the paradox dissolves.
     I don’t know whether this line of thought makes sense. [Note 21 Dec 2008: after some rewriting, it seems to me there’s a contradiction in it. Needs more work, but the contradiction may be fundamental.] Nor do I know whether Russell or someone else has explored the consequences of forbidding that a set may be its own subset. It does not, as far as I can tell, forbid that a subset may of the same cardinality as the set (as is the case with infinite sets).
      Footnote 1: Intersections and unions of sets will be of mixed type. Eg, if we define L(l), l=living, then intersection K of L and H will be K(h, l). Etc.
      Footnote 2: The notation needs to be worked out some more. Let H<1 n="">(h) be a set of n elements of type h. Then some subset of it would be H(h).
      Footnote 3: It’s probably all nonsense.
     Good book. **1/2 (2008)