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.

Theories of Everything and Godel's theorem

Does Godel's incompleteness theorem entail that the physicist's dream of a Theory of Everything (ToE) is impossible? It's a question which, curiously, has received scant attention in the philosophy of physics literature.

To understand the question, first we'll need to introduce some concepts from mathematical logic: A theory T is a set of sentences, in some language, which is closed under logical implication. In other words, any sentence which can be derived from a subset of the sentences in a theory, is itself a sentence in the theory. A model M for a theory T is an interpretation of the variables, predicates, relations and operations of the langauge in which that theory is expressed, which renders each sentence in the theory as true. Theories generally have many different models: for example, each different vector space is a model for the theory of vector spaces, and each different group is a model for the theory of groups. Conversely, given any model, there is a theory Th(M) which consists of the sentences which are true in the model M.

Now, a theory T is defined to be complete if for any sentence s, either s or Not(s) belongs to T. A theory T is defined to be decidable if there is an effective procedure of deciding whether any given sentence s belongs to T, (where an 'effective procedure' is generally defined to be a finitely-specifiable sequence of algorithmic steps). A theory is axiomatizable if there is a decidable set of sentences in the theory, whose closure under logical implication equals the entire theory.

It transpires that the theory of arithmetic (technically, Peano arithmetic) is both incomplete and undecidable. Moreover, whilst Peano arithmetic is axiomatizable, there is a particular model of Peano arithmetic, whose theory is typically referred to as Number theory, which Godel demonstrated to be undecidable and non-axiomatizable. Godel obtained sentences s, which are true in the model, but which cannot be proven from the theory of the model. These sentences are of the self-referential form, s = 'I am not provable from A', where A is a subset of sentences in the theory.

Whilst the application of mathematics to the physical world may be fairly untroubled by the difficulties of self-referential statements, undecidable statements which are free from self-reference have been found in various branches of mathematics. For example, it has been established that there is no general means of proving whether or not a pair of 'triangulated' 4-dimensional manifolds are homeomorphic (topologically identical).

Any theory which includes Number theory will be undecidable, hence if a final Theory of Everything includes Number theory, then the final theory will also be undecidable. The use of Number theory is fairly pervasive in mathematical physics, hence, at first sight, this appears to be highly damaging to the prospects for a final Theory of Everything in physics.

However, it is still conceivable that a final Theory of Everything might not include Number theory, and in this case, a final Theory of Everything could be both complete and decidable. In addition, even if a final Theory of Everything is incomplete and undecidable, it is the models M of a theory which purport to represent physical reality, and whilst the theory of a model Th(M) may be undecidable, it is guaranteed to be complete. That is, every sentence in the language of the theory will either belong or not belong to Th(M).

Lee Smolin and the multiverse

Lee Smolin argues in Physics World against the notion that there exists a multiverse of timeless universes. Smolin believes that the need to invoke a multiverse is rooted in the dichotomy between laws and initial conditions in existing theoretical physics, and suggests moving beyond this paradigm.

A choice of initial conditions, however, is merely one of the means by which particular solutions to the laws of physics are identified. More generally, there are boundary conditions, and free parameters in the equations, which have no special relationship to the nature of time. Each theory in physics represents (a part of) the physical universe by a mathematical structure; the laws associated with that theory select a particular sub-class of models with that structure; and the application of a theory to explain or predict a particular empirical phenomenon requires the selection of a particular solution, i.e., a particular model. The choice of initial conditions, or boundary conditions, or the choice of particular values for the free parameters in the equations, is simply a way of picking out a particular model of a mathematical structure. For example, in general relativity, the structure is that of a 4-dimensional Lorentzian manifold, the Einstein field equations select a sub-class of all the possible 4-dimensional Lorentzian manifolds, and the choice of boundary conditions or initial conditions selects a particular 4-dimensional Lorentzian manifold within that sub-class.

As a consequence, any 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. Irrespective of whether a future theory abolishes the dictotomy between laws and initial conditions, the application of that theory will require a means of identifying particular models of the mathematical structure selected by the theory. If there is only one physical universe, as Smolin claims, then the problem of contingency will remain: why does this particular model exist and not any one of the other possibilities? The invocation of a multiverse solves the problem of contingency by postulating that all the possible models physically exist.

The creation of Brown-man

2:30am.

Gordon Brown stared with dour misery at the digital bedside clock. The red digits burned with laser-like intensity in the dark, third-floor bedroom of Number 10. Gordon noticed, not for the first time, that the digits were actually composed of hexagonal and trapezoidal lozenges, and wondered with fresh irritation why they weren't simply rectangular.

Gordon knew that he wasn't going to get any sleep tonight. He could literally feel his spleen burning in anger, and it was fuelling his brain stem, which wailed like a banshee in his head, a venegeful turbine spinning at full throttle. Ploys, plots and revenge scenarios exploded like fireworks across his mind's eye.

Gordon levered himself onto one elbow, and glanced at Sarah. Her large-boned frame lay motionless across the bed, her breath shallow and regular.

Gordon knew that something had to be done, and it had to be radical, much as he despised that Blairite term.

3:00am

At the wheel of his Jaguar, still dressed in pyjamas, Gordon hurtled down the A12. Chelmsford, Colchester, Ipswich. The road signs zoomed past, but Gordon's focus was fixed in the middle-distance. He knew now what must be done, and would not waver. The speedometer hit 100mph. 105. 110. 115. Gordon's unblinking left eye glinted under the baleful sodium roadlights, hanging like spectres over the central reservation.

5:00am

Sizewell B. Gordon lowered himself with grim determination into the cooling waters of the reactor, and bathed in the rejuvenating and transformative blue glow of the Cerenkov radiation.

5:30am

In a Suffolk forest glade, Gordon knelt, head bowed, on a spongey patch of moss. Gnarled roots and trunks and over-hanging boughs surrounded the Prime Minister. Gordon's head and torso throbbed with an opalescent glow. He twisted and screamed in agony and elation as every cell in his body transformed itself, re-arranging its DNA, creating trillions of super-enzymes and super-proteins. The calcium in Gordon's bones transmogrified into an allotropic titanium-plutonium alloy; his nervous system mutated into a broadband fibre-optic TCP/IP network; his muscles became a carbonfibre-reinforced weave; his organs became high-capacity chemical processing plants.

Gordon raised his head and grinned manically. Standing now upon the mossy platform, his body began to pulse with fetid energy. Blue sparks discharged to the ground, and a growing aura surrounded the Prime Minister. "Behold," he proclaimed with a sonic shock-wave, "I'm a pretty straight sorta guy as well!"

And with that, a blinding flash of blue light exploded from his frame, and a luminous wave of purple coruscation propagated outwards through the forest. In its wake, as the first shafts of dawn sunlight penetrated the glade, the wood was transformed into a jewel-encrusted kaleidoscope of light and colour. The facets of a billion jewels, in a million different shades, shimmered and sparkled and glimmered on trunk and bough; the forest floor became a green velvet carpet; and leafs made of gold and silver shone upon the bejewelled branches above.

The Prime Minister marched out of the clearing. The world needed saving, and Brown-man was here to do it.

Simon Singh and Mr Justice Eady

Mr Justice Eady is, apparently, a "quietly-spoken, shy and precise figure." He is also the most senior libel court judge in the country. Mr Eady presided over Max Mosley's successful libel action against The News of the World last year, and his strong anti-media line is apparently motivated by an intrusion of privacy suffered by the actor Gordon Kaye in 1990. Perhaps he's a big 'Allo 'Allo fan.

Some readers may be aware that the well-known science writer Simon Singh was recently judged to have libelled the Bristish Chiropractic Association (BCA). Although The Times newspaper appears not to have reported the story, readers of other publications will be aware that the purported libel was Singh's claim, in an article for The Guardian, that the BCA "happily promotes bogus treatments". Whilst many people might interpret this claim to mean that the BCA promotes false treatments, the judge presiding over the case chose to provide his own interpretation, in which the "natural and ordinary meaning" of the assertion was that the BCA were being deliberately dishonest.

And the name of the judge in question? Pay attention, for I shall say this only once: Mr Justice Eady.