On Mon, May 23, 2005 at 04:00:39AM +0100, Patrick Leahy wrote:
> 
> 
> 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".
> 

A finite set of axioms describing the reals does not completely
specify the real numbers, unless they are inconsistent. I'm sure you've
heard it before. The system the axioms do describe can be modelled by a
countable system as well. You could say that it describes describable
functions over describable numbers. It may even be the case that you
only have computable functions over computable numbers, and that
describable, uncomputable things cannot be captured by finite
axiomatic systems, but I'm not sure. Juergen Schmidhuber knows more
about such things.

What I would argue is what use are undescribable things in the
plenitude? Hence my interpretation of Tegmark's assertion is of finite
axiomatic systems, not all mathematic things.

Cheers

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