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" ismuch 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 beconvincing. As I said often, physical reality cannot be a mathematicalreality *among other*. The relation is more subtle both with or without thecomp 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 notthe whole.

`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 evenexpressible 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".`

Paddy Leahy