On Mon, 23 May 2005, Russell Standish wrote:
I think most of us concluded that Tegmark's thesis is somewhat ambiguous. One "interpretation" of it that both myself and Bruno tend to make is that it is the set of finite axiomatic systems (finite sets of axioms, and recusively enumerated theorems). Thus, for example, the system where the continuum hypothesis is true is a distinct mathematical system from one where it is false. Such a collection can be shown to be a subset of the set of descriptions (what I call the Schmidhuber ensemble in my paper), and has some fairly natural measures associated with it. As such, the arguments I make in "Why Occam's razor paper" apply just as much to Tegmark's ensemble as Schmidhuber's.
Hmm, my lack of a pure maths background may be getting me into trouble here. What about real numbers? Do you need an infinite axiomatic system to handle them? Because it seems to me that your ensemble of digital strings is too small (wrong cardinality?) to handle the set of functions of real variables over the continuum. Certainly this is explicit in Schmidhuber's 1998 paper. Not that I would insist that our universe really does involve real numbers, but I'm pretty sure that Tegmark would not be happy to exclude them from his "all of mathematics".
Conversely, if you wish to stand on the phrase "all of mathematics exists" then you will have trouble defining exactly what that means, let alone defining a measure.
I don't wish to, but this concept has been repeated by Tegmark in several well publicised articles (e.g. the Scientific American one). Again, lack of mathematical background forbids me from making definitive claims, but I suspect that it could be proved impossible even to define a measure over *all* self-consistent mathematical concepts. In which case Lewis was right and Tegmark's "level 4 multiverse" is essentially content-free, from the point of view of a physicist (as opposed to a logician).