1 1 Jouko Väänänen How complicated can structures be? NAW 5/9 nr. 2 June 2008 117 Jouko Väänänen University of Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam The Netherlands [email protected] How complicated can structures be? Is there a measure of how ’close’ non-isomorphic mathematical structures are? Jouko functions f : N → N endowed with the topolo- Väänänen, professor of logic at the Universities of Amsterdam and Helsinki, shows how con- gy of pointwise convergence, where N is given temporary logic, in particular set theory and model theory, provides a vehicle for a meaningful the discrete topology. discussion of this question. As the journey proceeds, we accelerate to higher and higher We can now consider the orbit of an arbi- cardinalities, so fasten your seatbelts. trary countable structure under all permuta- tions of N and ask how complex this set is By a structure we mean a set endowed with a eyes of the difficult uncountable structures. in the topological space N. If the orbit is a finite number of relations, functions and con- New ideas are needed here and a lot of work closed set in this topology, we should think of stants. Examples of structures are groups, lies ahead. Finally we tie the uncountable the structure as an uncomplicated one. This fields, ordered sets and graphs. Such struc- case to stability theory, a recent trend in mod- is because the orbit being closed means, in tures can have great complexity and indeed el theory. It turns out that stability theory and view of the definition of the topology, that the this is a good reason to concentrate on the the topological approach proposed here give finite parts of the structure completely deter- less complicated ones and to try to make similar suggestions as to what is complicated mine the whole structure, as is easily seen to some sense of them. In this article we walk and what is not. be the case in the graph of the picture: the less obvious and perhaps less appealing For unexplained set theoretical concepts trail of delving more deeply into more and refer to [5]. more complicated structures. We raise the question of how we can make sense of the Finite structures statement that we have found an extremely Let us start with finite structures. The fa- complicated structure? This is typical of the mous P=NP question, one of the Clay Insti- kind of question investigated in mathematical tute Millennium Questions, asks if we can logic. The guiding result of mathematical log- decide in polynomial time whether a given ic is the Incompleteness Theorem of Gödel, finite graph is 3-colourable. Should the an- which says that the logical structure of num- swer to the P=NP question be negative, as is A structure may be quite innocuous even if ber theory is so complicated that it cannot be expected, we will have a sequence of some the orbit is not closed. For example, the or- effectively axiomatized in its entirety. In other rather complicated graphs, for which no algo- bit of the ordered set of the rationals is not words, the theory is non-recursive, i.e. there rithm, running in polynomial time in the size closed because as far as the finite parts are is no Turing machine that could tell whether of the graph, can decide whether the graph is concerned it cannot be distinguished from the a sentence of number theory is true or not. 3-colourable or not. order type of the integers. While not closed, A contrasting and pivotal result of logic from The problem of whether the isomorphism the orbit of the rationals is of the form the same period is Alfred Tarski’s result that of two finite structures can be solved in poly- the field of real numbers (or the field of com- nomial time is a famous open problem of \ [ Fn,m, (1) plex numbers) can be completely and effec- complexity theory. It is particularly famous n m tively axiomatized and is indeed recursive in because it is not known whether it is NP- the sense that there is a Turing machine that complete either; it may be strictly between where each Fn,m is closed. This is a con- decides whether a given statement about the P and NP. sequence of the fact that the density of the plus and times of real (or complex) numbers order, as well as not having endpoints, can is true or not. Countable structures be expressed in the form ‘for all ... exists ...’, We start with the extremely interesting sit- What about countably infinite structures? We and these two properties completely deter- uation concerning attempts to classify finite should not distinguish between isomorphic mine the structure among countable struc- models. We then move to the more estab- structures. So let us assume the universe tures. lished case of countable structures. Sweep- of our countable structures is the set N of When the number of alternating intersec- ing results exist here and this case is very natural numbers. After a little bit of coding, tions and unions increases in the formula (1), much the focus of current research. Then we such countable structures can be thought of even to the transfinite, we end up with the turn our faces to the wind and stare into the as points in the topological space N of all hierarchy of Borel sets 1 1 2 2 118 NAW 5/9 nr. 2 June 2008 How complicated can structures be? Jouko Väänänen Illustration: Ryu Tajiri 2 2 3 3 Jouko Väänänen How complicated can structures be? NAW 5/9 nr. 2 June 2008 119 0 • G0 = open sets, F0 = closed sets morphic with L . So in some sense L is a 0 • Gα+1 = countable unions of sets from Fα hair’s breadth away from being L . The fact • Fα+1 = countable intersections of sets from that L is complicated is related to exactly this Gα kind of phenomenon, to being an iota away S S • Gν = α<ν Gα, Fν = α<ν Fα, if ν is a from another, non-isomorphic structure. limit ordinal When we set our foot on the path of look- named after Émile Borel (1871–1956), a ing at uncountable structures through the French mathematician. lens of their countable parts, the first rest- So the philosophy is now that the further ing spot is bound to be the class of structures the orbit is from being a closed set the more that can be expressed as an increasing union complicated the structure is. We can go up of countable substructures or, equivalently, the Borel hierarchy and find structures on all structures of cardinality ℵ1, the first uncount- levels Fα ∪Gα. By a deep result of Dana Scott able cardinal. Now the alarm bells start to [9] every orbit is on some level of the Borel ring! We do not know whether the real num- hierarchy, although a priori the orbits are just bers, the complex numbers, Euclidean space, analytic sets, i.e. continuous images of closed Banach spaces, etc, have this property. sets. The question of whether the set R of real The levels Fα ∪ Gα of the Borel hierarchy numbers is an increasing union of countable are calibrated by countable ordinals α. Or- sets is known as the Continuum Hypothesis Émile Borel (1871–1956), a French math- bits of familiar structures such as (N, +, ·, 0, 1), (CH). So in order to include those structures the field of rational numbers, the Random in this discussion we have to assume CH. In ematician and politician who has many Graph, the free Abelian group of countably fact, most of the currently known results in theorems named after him; there is many generators, and any vector space (over this direction assume CH anyway. But there even a Borel crater on the moon in the Mare Serenitatis. Borel, together with Q) of countable dimension are all on one of is a whole family of structures that are by their the lowest infinite levels of the Borel hierar- very definition increasing unions of countable Lebesgue and Baire, is also known as chy. On the other hand the orbit of any suf- structures, and this family is closed under var- a representative of semi-intuitionism, an ficiently closed countable ordinal (α, <) is on ious algebraic operations but not under infi- alternative approach to constructivism along with Brouwer’s intuitionism. The level α, i.e. in the set Fα but not in any Fβ ∪Gβ nite products, unless we assume CH. An ex- former maintained that set theory should for β < α (what is needed is that α is such that ample is the order-type (ω1, <) and the nu- β < α implies ωβ < α). Such structures of merous structures built around it, such as the be limited to definable sets, anticipat- ing descriptive set theory, while the latter high level can be constructed for e.g. Abelian free Abelian group on ℵ1 generators. launched a criticism of the Law of Exclud- groups. This basic setup has led recently to a Models of cardinality ℵ1 can be thought ed Middle, leading to modern intuitionis- rich theory of Borel equivalence relations on of as points in the space N1 of functions tic logic and constructive mathematics. Polish spaces [1]. f : ω1 → ω1 endowed with the topology of pointwise convergence, that is, a neigh- Uncountable structures bourhood of a point f ∈ N1 is of the form What if we have an uncountable structure and N(f , X) = {g ∈ N1 : ∀x ∈ X(g(x) = f (x))} approximation is more complex.