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. After all, we we want to measure its complexity and the where X is countable. So the orbit of a struc- have to approximate an uncountable object
degree to which a given structure is close to ture is a closed set essentially if the countable and we cannot approximate f ∈ N1 merely being isomorphic to it? After all, the most parts of the structure completely determine by its finite initial segments. Roughly for the
important mathematical structures, such as it. The orbit of the free abelian groep F(ℵ1) same reason, the approximations are scaled the fields of real numbers and complex num- on ℵ1 is not at all closed. There are so-called by bounded trees, i.e. trees with no uncount- bers, Euclidean spaces, Banach spaces, etc., almost free Abelian groups, every countable able branches, rather than ordinals. are all uncountable. In the light of our experi- subgroup of which is free but which are not The passage from well-founded trees to ence with countable structures, it seems natu- free themselves. So the question to ask is bounded trees brings with it two major prob- ral to consider structures that are determined not only what the countable substructures are lems. The first is the problem of the ordering by their countable parts as uncomplicated. but also how they sit inside the structure. To of the class of all such trees. The problem of Consider the ordered set L = (R, <) of all see how complicated the group F(ℵ1) is let us the ordering of the trees is the following. In real numbers. If we only look at countable define the Borel hierarchy in N1. analogy with ordinals and well-founded trees 0 sub-orders, this is no different from the or- The class of Borel sets of the space N1 we write T ≤ T if there is a strict tree-order 0 dered set L = (R \{0}, <) of the non-zero is the smallest class of sets containing the preserving (but not necessarily one-to-one) real numbers — although L and L0 are not open sets and closed under complements mapping from T to T 0. For well-founded trees
isomorphic as the first is a complete order and unions of length ω1. A set is analytic this quasi-order is connected, i.e. any two and the second is not. In fact, L is quite a if it is a continuous image of a closed subset well-founded trees are comparable by ≤. For
complicated structure albeit not by any means of N1. Orbits of structures of cardinality ℵ1 non-well-founded trees this need not be the among the most complicated. One example are, a priori, analytic, but are they Borel? case [4, 13]. This is on the one hand a short- of the peculiar properties of L is the follow- When we carry out the same topological coming of the whole approach, as it means
ing. If we add a new real to the universe by analysis of models of cardinality ℵ1, as we that some structures are incomparable as to Cohen’s method of forcing, L becomes iso- did with countable models, the notion of an their complexity. On the other hand it has un-
3 3 4 4
120 NAW 5/9 nr. 2 June 2008 How complicated can structures be? Jouko Väänänen
earthed a rich theory of trees, and whatever the Luzin Separation Principle fails. There are theoretic sense described in this article [3]. To progress we can make in this direction is di- disjoint analytic sets that cannot be separat- measure the height of the complexity one us- rectly reflected in our ability to measure how ed by a Borel set [11]. This further emphasizes es bounded trees and it turns out that under close uncountable structures can be to each how the difference between the countable the stated assumptions one can go beyond other in the given topology. One puzzle that and the uncountable is reflected in the topol- any bounded tree. Work in this direction is
has arisen in this connection is the existence ogy of N and N1, and thereby in the classifica- very much underway. of a Canary tree. This name is due to the fol- tion of countable versus uncountable models. lowing special role of Canary trees. If any sta- Conclusion
tionary subset of ω1 is killed by forcing with- Stability theory The study of the complexity of uncountable out adding reals, then the Canary tree gets In modern model theory there is an alterna- structures is an interdisciplinary subject. We a long branch. Summing up, if a stationary tive approach to the problem of classifying need to develop set theory, and especially set is poisoned somewhere then the Canary structures, namely stability theory [8]. The the theory of trees, in order to have a good tree warns us by expiring. The existence of difference is that stability theory tries to clas- measure of the complexity of uncountable Canary trees cannot be decided on the basis sify complete first order theories rather than models. At the same time, we have to de- of ZFC or even CH alone [7]. However, Canary structures. However, the message of stability velop model theory, and especially stability
trees are intimately related to the complexity theory is that all models of size ℵ1 of theories theory, in order to distinguish important di- of some canonical structures. It can be shown satisfying a combination of certain stability viding lines between simple and complicated that there is a Canary tree if and only if the or- conditions (superstable, NDOP, DOTOP) are structures. Both set theory and model the-
bit of the free Abelian group of ℵ1 generators rather ‘uncomplicated’ in the sense that their ory suggest that we should look for an an- is analytic co-analytic [6]. In the space N, isomorphism can be expressed in terms of swer in the direction of long games. In mod- analytic co-analytic sets are Borel but in the a determined game (called the Ehrenfeucht- el theory the relevant games are known as
space N1, the situation is more complicated. Fraïssé-game) of length ω [10]. Naturally, Ehrenfeucht-Fraïssé games. In set theory the The most promising attempt to bring order in- such structures may be very complicated in corresponding games are related to the so- to the chaos of bounded trees is the approach other ways. The point of stability theory is that called stationary sets. The results referred of Todorcevic [12] under the assumption of the in such structures one can define a kind of ge- to above connect the two games and thereby so-called Proper Forcing Axiom (PFA). This ax- ometry that enables one to classify the struc- tie a knot connecting set theory and model iom says, very roughly speaking, that the uni- ture in terms of dimension-like invariants. On theory. When we look into the deep eyes of verse is invariant under changes imposed by the other hand, theories failing to satisfy such the uncountable structures, we are perhaps a certain restricted form of Cohen’s concept stability conditions are bound to have models starting to see there some compassion for our
of forcing. Another approach is to restrict to of cardinality ℵ1 that are extremely complicat- modest advances, our budding infinite trees, sufficiently definable trees and thereby avoid ed. This is Shelah’s ‘Main Gap’ [10]. For exam- our courageous appeals to stability and our the incomparability problem [2]. ple, assuming CH and if there are no Canary resolve to play the game to the end. k Another new feature that arises in the trees, such theories have models of cardinal-
study of N1 is the fact that if we assume CH, ity ℵ1 with high complexity in the definability
References 1 Howard Becker and Alexander S. Kechris, The 5 Kenneth Kunen,Set theory, volume 102 of Stud- ory of Models (Proc. 1963 Internat. Sympos. descriptive set theory of Polish group actions, ies in Logic and the Foundations of Mathemat- Berkeley), pages 329–341. North-Holland, Am- volume 232 of London Mathematical Soci- ics. North-Holland Publishing Co., Amsterdam, sterdam, 1965. ety Lecture Note Series. Cambridge University 1983. An introduction to independence proofs. 10 S. Shelah, Classification theory and the number Press, Cambridge, 1996. Reprint of the 1980 original. of nonisomorphic models, volume 92 of Stud- 2 Ilijas Farah and Paul B. Larson, ‘Absoluteness 6 Alan Mekler and Jouko Väänänen, ‘Trees and ies in Logic and the Foundations of Mathemat- for universally Baire sets and the uncountable’ 1 ω1 ics. North-Holland Publishing Co., Amsterdam, Π1-subsets of ω1’, J. Symbolic Logic, I. In Set theory: recent trends and applications, 58(3):1052–1070, 1993. second edition, 1990. volume 17 of Quad. Mat., pages 47–92. Dept. 11 Saharon Shelah and Jouko Väänänen, ‘Station- Math., Seconda Univ. Napoli, Caserta, 2006. 7 Alan H. Mekler and Saharon Shelah, ‘The canary tree’, Canad. Math. Bull., 36(2):209–215, 1993. ary sets and infinitary logic’, J. Symbolic Logic, 3 Tapani Hyttinen and Heikki Tuuri, ‘Construct- 65(3):1311–1320, 2000. ing strongly equivalent nonisomorphic models 8 Anand Pillay, Geometric stability theory, vol- 12 Stevo Todorceviˇ c,´ ‘Lipschitz maps on trees’, J. for unstable theories’, Ann. Pure Appl. Logic, ume 32 of Oxford Logic Guides, The Clarendon Inst. Math. Jussieu, 6(3):527–556, 2007. 52(3):203–248, 1991. Press Oxford University Press, New York, 1996. Oxford Science Publications. 13 Stevo Todorceviˇ c´ and Jouko Väänänen, ‘Trees 4 Tapani Hyttinen and Jouko Väänänen, ‘On Scott and Ehrenfeucht-Fraïssé games’, Ann. Pure Ap- and Karp trees of uncountable models’, J. Sym- 9 Dana Scott, ‘Logic with denumerably long for- pl. Logic, 100(1-3):69–97, 1999. bolic Logic, 55(3):897–908, 1990. mulas and finite strings of quantifiers’, In The-
4 4