Now I'm really confused!
I took Russell to mean that real numbers are excluded from his system
because they require an infinite number of axioms. In which case his
system is really quite different from Tegmark's.
But if Bruno is correct and reals only need a finite number of axioms,
then surely Russell is wrong to imply that real-number universes are
covered by his system.
Sure, they can be modelled to any finite degree of precision, but that is
not the same thing as actually being included (which requires infinite
precision). For instance, Duhem pointed out that you can devise a
Newtonian dynamical system where a particle will go to infinity if its
starting point is an irrational number, but execute closed orbits if its
starting point is rational.
On Mon, 23 May 2005, Bruno Marchal wrote (among other things):
Le 23-mai-05, à 06:09, Russell Standish a écrit :
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).
Since Tegmark defines "mathematical structures" as existing if
self-consistent (following Hilbert), how can his concept be inconsistent?
But there may be an inconsistency in (i) asserting the identity of
isomorphic systems and (ii) claiming that a measure exists, especially if
you try both at once.
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 list.
Category theory and logic provides tools for defining big structure, but not
As I understand it, this is because "the whole" is unquantifiably big,
i.e. outside even the heirarchy of cardinals. Correct?
The David Lewis problem mentionned recently is not even
expressible in Tegmark framework.
It might be illuminating if you could explain why not. On the face of it,
it fits in perfectly well, viz: for any given lawful universe, there are
infinitely many others in which as well as the observable phenomena there
exist non-observable "epiphenomenal rubbish". The only difference from the
White Rabbit problem is the specification that the rubbish be strictly
non-observable. As a physicist, my reaction is that it is then irrelevant
so who cares? But this can be fixed by making the rubbish perceptible but
mostly harmless, i.e. "White Rabbits".