Tuesday, June 23, 2009

Mathematical logic and multiverses

The concepts of mathematical logic, introduced to explain Godel's theorem, can also be exploited to shed further light on the question of multiverses in mathematical physics.

Recall that any physical theory whose domain extends to the entire universe, (i.e. any cosmological theory), has a multiverse associated with it: namely, the class of all models of that theory. Both complete and incomplete theories are capable of generating such multiverses. The class of models of a complete theory will be mutually non-isomorphic, but they will nevertheless be elementarily equivalent. Two models of a theory are defined to be elementarily equivalent if they share the same truth-values for all the sentences of the language. Whilst isomorphic models must be elementarily equivalent, there is no need for elementarily equivalent models to be isomorphic. Recalling that a complete theory T is one in which any sentence s, or its negation Not(s), belongs to the theory T, it follows that every model of a complete theory must be elementarily equivalent.

Alternatively, if a theory is such that there are sentences which are true in some models but not in others, then that theory must be incomplete. In this case, the models of the theory will be mutually non-isomorphic and elementarily inequivalent.

Hence, mathematical logic suggests that the application of mathematical physics to the universe as a whole can generate two different types of multiverse: classes of non-isomorphic but elementarily equivalent models; and classes of non-isomorphic and elementarily inequivalent models.

The question then arises: are there any conditions under which a theory has only one model, up to isomorphism? In other words, are there conditions under which a theory doesn't generate a multiverse, and the problem of contingency ('Why this universe and not some other?') is eliminated?

A corollary of the Upward Lowenheim-Skolem theorem provides an answer to this. The latter entails that any theory which has a model of any infinite cardinality, will have models of all infinite cardinalities. Models of different cardinality obviously cannot be isomorphic, hence any theory, complete or incomplete, which has at least one model of infinite cardinality, will have a multiverse associated with. (In the case of a complete theory, the models of different cardinality will be elementarily equivalent, even if they are non-isomorphic). Needless to say, general relativity has models which employ the cardinality of the continuum, hence general relativity will possess models of every cardinality.

For a theory of mathematical physics to have only one possible model, it must have only a finite model. A Theory of Everything must have a unique finite model if the problem of contingency, and the potential existence of a multiverse is to be eliminated.


Doug Hudson said...

Brilliant stuff Gordon. Is this original material? If so, you should put it on a solid mathematical grounding and publish it!

Gordon McCabe said...

Cheers Doug!

Coincidentally, I've just started the process of writing this into a paper for submission to one of the journals. The mathematical logic is well-known to mathematical logicians, but cosmologists appear to be largely oblivious of it. Which, in theory, is exactly where philosophy of physics should have a part to play...

Anonymous said...

A Theory of Everything must have a unique finite model if the problem of contingency, and the potential existence of a multiverse is to be eliminated."

I believe an model is suppose to be an interpretation of a formal theory. So, for a complete cosmological formal theory T, Every theorm derivable in T has an interpretation, descriabled by a model M.

If we can consider all the possible ways things could be, or all the possible intrepretation for T, then this forms an essemble of models E. Each derivable statements in T correspondes to some elements in E.

So, for a practical example. I can think of a cosmological theory T as being this TOE. T would be either a single equation or multiple equations. In both case, there would be constants & initial conditions. One can imagine varying the constants, parameters in T to form a multiverse.

If What i am saying make sense, then for a theory T, there is more than 1 model, and if we can consider a set of models as forming an multiverse, then multiverse is implied for every cosmological theory.