At 11:34 -0700 23/09/2002, Hal Finney wrote:
>I have gone back to Tegmark's paper, which is discussed informally
>at http://www.hep.upenn.edu/~max/toe.html and linked from
>http://arXiv.org/abs/gr-qc/9704009.
>
>I see that Russell is right, and that Tegmark does identify mathematical
>structures with formal systems. His chart at the first link above shows
>"Formal Systems" as the foundation for all mathematical structures.
>And the discussion in his paper is entirely in terms of formal systems
>and their properties. He does not seem to consider the implications if
>any of Godel's theorem.
>
>I still think it is an interesting question whether this is the only
>possible perspective, or whether one could meaningfully think of an
>ensemble theory built on mathematical structures considered in a more
>intuitionist and Platonic model, where they have existence that is more
>fundamental than what we capture in our axioms. Even if this is not
>what Tegmark had in mind, it is an alternative ensemble theory that is
>worth considering.

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... and comp leads naturally toward such an alternative ensemble theory.
You can look again at Tegmark's Chart, substitute "formal system" by
machines, all the rest are machine dreams. But comp constraints forces us
not only to put a measure on those dreams, but to extract the (1)-measure
from Godel-Lob theorems, actually from the whole logic of self-reference.
(This is what I have partially done, not exactly in those terms, because
it was a long time before Tegmark wrote is paper).
Bruno