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 the 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 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".

Paddy Leahy

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