Jonathan Colvin writes:
> That's rather the million-dollar question, isn't it? But isn't the
> multiverse limited in what axioms or predicates can be assumed? For
> instance, can't we assume that in no universe in Platonia can (P AND ~P) be
> an axiom or predicate?
No, I'd say that you could indeed have a mathematical object which had P
AND ~P as one of its axioms. The problem is that from a contradiction,
you can deduce any proposition. Therefore this mathematical system can
prove all well formed strings as theorems.
As Russell mentioned the other day, "everything" is just the other side of
the coin from "nothing". A system that proves everything is essentially
equivalent to a system that proves nothing. So any system based on P AND
~P has essentially no internal structure and is essentially equivalent
to the empty set.
One point is that we need to distinguish the tools we use to analyze
and understand the structure of a Tegmarkian multiverse from the
mathematical objects which are said to make up the multiverse itself.
We use logic and other tools to understand the nature of mathematics;
but mathematical objects themselves can be based on nonstandard "logics"
and have exotic structures. There are an infinite number of formal
systems that have nothing to do with logic at all. Formal systems are
just string-manipulation engines and only a few of them have the axioms
of logic as a basis. Yet they can all be considered mathematical objects
and some of them might be said to be universes containing observers.