Regarding the nature of Tegmark's mathematical objects, I found some
old discussion on the list, a debate between me and Russell Standish,
in which Russell argued that Tegmark's objects should be understood as
formal systems, while I claimed that they should be seen more as pure
Platonic objects which can only be approximated via axiomatization.

The discussions can generally be found at
http://www.escribe.com/science/theory/index.html?by=Date&n=25 under
the title "Tegmark's TOE & Cantor's Absolute Infinity".  In particular

http://www.escribe.com/science/theory/m4034.html
http://www.escribe.com/science/theory/m4045.html
http://www.escribe.com/science/theory/m4048.html

In this last message I write:

> 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.

Note, Tegmark's paper has moved to
http://space.mit.edu/home/tegmark/toe_frames.html .

See also http://www.escribe.com/science/theory/m4038.html where Wei Dai
argues that even unaxiomatizable mathematical objects, even infinite
objects that are too big to be sets, can have a measure meaningfully
applied to them.  However I do not know enough math to fully understand
his proposal.

Hal Finney

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