Perhaps I can throw in a few thoughts here, partly in the hope I may learn something from possible replies (or lack thereof!).

----- Original Message ----- From: Patrick Leahy <[EMAIL PROTECTED]> Sent: 23 May 2005 00:03 . . > A very similar argument ("rubbish universes") was put forward long ago > against David Lewis's modal realism, and is discussed in his "On the > plurality of worlds". As I understand it, Lewis's defence was that there > is no "measure" in his concept of "possible worlds", so it is not > meaningful to make statements about which kinds of universe are "more > likely" (given that there is an infinity of both lawful and law-like > worlds). This is not a defense which Tegmark can make, since he does > require a measure (to give his thesis some anthropic content). I don't understand this last sentence - why couldn't he use the 'Lewisian defence' if he wanted - it is the Anthropic Principle (or just logic) that necessitates SAS's (in a many worlds context): our existence in a world that is suitable for us is independent of the uncountability or otherwise of the sets of suitable and unsuitable worlds, it seems to me. (Granted he does use the 'm' word in talking about level 4 (and other level) universes, but I am asking why he needs it to provide 'anthropic content'.) There are hints that it may be worth exploring fundamentally different approaches to the White Rabbit problem when we consider that for Cantor the set of all integers is the same 'size' as that of all the evens (not too good on its own for deciding whether a randomly selected integer is likely to come out odd or even); similarly for comparing the set of all reals between 0 and 1000, and between 0 and 1. The standard response to this is that one *cannot* select a real (or integer) in such circumstances - but in the case of many worlds we *do* have a selection (the one we are in now), so maybe there is more to be said than that of applying the Cantor approach to real worlds, and also on random selection. I use the simple 'limit to infinity' approach to provide a potential solution to the WR problem (see appendix of http://www.physica.freeserve.co.uk/pa01.htm) - Russell's paper is not too dissimilar in this area, I think. This approach seems to cover at least the 'countable' region (in Cantorian terms), and also addresses the above problems (ie odd/even type questions etc). The key point in my philosophy paper is that it is mathematics (and/or information theory) that is more likely to map the objective distribution of types of worlds, compared to the particular anthropic intuition that is implied by the WR challenge. A final musing on finite formal systems: I have always considered formal systems to be a provisional 'best guess' (or *maybe* 2nd best after the informational approach) for exploring the plenitude - but it occurs to me that non-finitary formal systems (which could inter alia encompass the reals) may match (say SAS-relevant) finite formal systems in simplicity terms, if the (infinite-length) axioms themselves could be algorithmically generated. This would lead to a kind of 'meta-formal-system' approach. Just a passing thought... Alastair