Le 23-mai-05, à 06:09, Russell Standish a écrit :
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
handle them? Because it seems to me that your ensemble of digital
is too small (wrong cardinality?) to handle the set of functions of
variables over the continuum. Certainly this is explicit in
1998 paper. Not that I would insist that our universe really does
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.
I guess you mean "natural numbers". By a theorem by Tarski there is a
sense to say that the real numbers are far much simpler than the
natural numbers (for example it has taken 300 years to prove Fermat
formula when the variables refer to natural numbers, but the same
formula is an easy exercice when the variable refers to real number).
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.
Then you get only the constructive real numbers. This is equivalent to
the total (defined everywhere) computable function from N to N. This is
a non recursively enumerable set. People can read the diagonalization
posts to the everything-list in my url, to understand better what
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.
It is a bit too ambiguous.
What I would argue is what use are undescribable things in the
I don't think we can escape them; provably so once the comp hyp is
Hence my interpretation of Tegmark's assertion is of finite
axiomatic systems, not all mathematic things.
I don't think Tegmark would agree. I agree with you that "the whole
math" is much too big (inconsistent).
The bigger problem with Tegmark is that he associates first person with
their third person description in a 1-1 way (like most aristotelian).
But then he should postulate non-comp, and explain the nature of that
association with a suitable theory of mind (which he does not really
It is mainly from a logician point of view that Tegmark can hardly be
convincing. As I said often, physical reality cannot be a mathematical
reality *among other*. The relation is more subtle both with or without
the comp hyp. I have discussed it at length a long time ago in this
Category theory and logic provides tools for defining big structure,
but not the whole. The David Lewis problem mentionned recently is not
even expressible in Tegmark framework. Schmidhuber takes the right
ontology, but then messed up completely the "mind-body" problem by
complete abstraction from the 1/3 distinction.
Tegmark do a sort of 1/3 distinction (the frog/bird views) but does not
take it sufficiently seriously.