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I. Coherent Sequences Stevo Todorcevic

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I. Coherent Sequences Stevo Todorcevic I. Coherent Sequences Stevo Todorcevic Contents 0 Introduction . 1 1 Walks in the space of countable ordinals . 3 2 The coherence of maximal weights . 9 3 Oscillations of traces . 21 4 Triangle inequalities . 26 5 Conditional weakly null sequences . 39 6 An ultra¯lter from a coherent sequence . 44 7 The trace and the square-bracket operation . 51 8 A square-bracket operation on a tree . 62 9 Special trees and Mahlo cardinals . 67 10 The weight function on successor cardinals . 73 11 The number of steps . 75 12 Square sequences . 80 13 The full lower trace of a square sequence . 90 14 Special square sequences . 96 15 Successors of regular cardinals . 98 16 Successors of singular cardinals . 109 17 The oscillation mapping . 117 18 The square-bracket operation . 122 19 Unbounded functions on successors of regular cardinals 129 20 Higher dimensions . 138 Bibliography . 152 Index . 160 0. Introduction A trans¯nite sequence C» ⊆ » (» < θ) of sets may have a number of 'coher- ence properties' and the purpose of this chapter is to study some of them as well as some of their uses. 'Coherence' usually means that the C»'s are cho- 1 2 I. Coherent Sequences sen in some canonical way, beyond the natural requirement that C» is closed and unbounded in » for all ». For example, choosing a canonical 'funda- mental sequence' of sets C» ⊆ » for » < "0 relying on the speci¯c properties of the Cantor normal form for ordinals below the ¯rst ordinal satisfying the equation x = !x is a basis for a number of important results in proof theory. In set theory, one is interested in longer sequences as well and usually has a di®erent perspective in applications, so one is naturally led to use some other tools beside the Cantor normal form. It turns out that the sets C» can not only be used as 'ladders' for climbing up in recursive constructions but also as tools for 'walking' from an ordinal to a smaller one. This notion of a 'walk' and the corresponding 'distance functions' constitute the main body of study in this chapter. We show that the resulting 'metric theory of ordinals' not only provides a uni¯ed approach to a number of classical problems in set theory but also has its own intrinsic interest. For example, from this theory one learns that the triangle inequality of an ultrametric e(®; γ) · maxfe(®; ¯); e(¯; γ)g has three versions, depending on the natural ordering between the ordi- nals ®, ¯ and γ, that are of a quite di®erent character and are occurring in quite di®erent places and constructions in set theory. For example, the most frequent occurrence is the case ® < ¯ < γ when the triangle inequal- ity becomes something that one can call 'transitivity' of e. Considerably more subtle is the case ® < γ < ¯ of this inequality. It is this case of the inequality that captures most of the coherence properties found in this article. Another thing one learns from this theory is the special role of the ¯rst uncountable ordinal in this theory. Any natural coherency requirement on the sets C» (» < θ) that one ¯nds in this theory is satis¯able in the case θ = !1. The ¯rst uncountable cardinal is the only cardinal on which the theory can be carried out without relying on additional axioms of set theory. The ¯rst uncountable cardinal is the place where the theory has its deepest applications as well as its most important open problems. This special role can perhaps be explained by the fact that many set-theoretical problems, especially those coming from other ¯elds of mathematics, are usually con- cerned only about the duality between the countable and the uncountable rather than some intricate relationship between two or more uncountable cardinalities. This is of course not to say that an intricate relationship be- tween two or more uncountable cardinalities may not be a pro¯table detour in the course of solving such a problem. In fact, this is one of the reasons for our attempt to develop the metric theory of ordinals without restricting ourselves only to the realm of countable ordinals. The chapter is organized as a discussion of four basic distance functions on ordinals, ½; ½0; ½1 and ½2, and the reader may choose to follow the anal- ysis of any of these functions in various contexts. The distance functions will naturally lead us to many other derived objects, most prominent of 1. Walks in the space of countable ordinals 3 which is the 'square-bracket operation' that gives us a way to transfer the quanti¯er 'for every unbounded set' to the quanti¯er 'for every closed and unbounded set'. This reduction of quanti¯ers has proven to be quite useful in constructions of various mathematical structures, some of which have been mentioned or reproduced here. I wish to thank Bernhard KÄonig,Piotr Koszmider, Justin Moore and Christine HÄartlfor help in the preparation of the manuscript. 1. Walks in the space of countable ordinals The space !1 of countable ordinals is by far the most interesting space considered in this chapter. There are many mathematical problems whose combinatorial essence can be reformulated as problems about !1, which is in some sense the smallest uncountable structure. What we mean by 'struc- ture' is !1 together with system C® (® < !1) of fundamental sequences, i.e. a system with the following two properties: (a) C®+1 = f®g, (b) C® is an unbounded subset of ® of order-type !, whenever ® is a countable limit ordinal > 0. Despite its simplicity, this structure can be used to derive virtually all known other structures that have been de¯ned so far on !1. There is a nat- ural recursive way of picking up the fundamental sequences C®, a recursion that refers to the Cantor normal form which works well for, say, ordinals 1 < "0 . For longer fundamental sequences one typically relies on some other principles of recursive de¯nitions and one typically works with fundamen- tal sequences with as few extra properties as possible. We shall see that the following assumption is what is frequently needed and will therefore be implicitly assumed whenever necessary: (c) if ® is a limit ordinal, then C® does not contain limit ordinals. 1.1 De¯nition. A step from a countable ordinal ¯ towards a smaller ordinal ® is the minimal point of C¯ that is ¸ ®. The cardinality of the set C¯ \ ®, or better to say the order-type of this set, is the weight of the step. 1.2 De¯nition. A walk (or a minimal walk ) from a countable ordinal ¯ to a smaller ordinal ® is the sequence ¯ = ¯0 > ¯1 > : : : > ¯n = ® such that for each i < n, the ordinal ¯i+1 is the step from ¯i towards ®. 1 One is tempted to believe that the recursion can be stretched all the way up to !1 and this is probably the way P.S. Alexandro® has found his famous Pressing Down Lemma (see [1] and [2, appendix]). 4 I. Coherent Sequences Analysis of this notion leads to several two-place functions on !1 that give a rich structure with many applications. So let us expose some of these functions. 2 <! 1.3 De¯nition. The full code of the walk is the function ½0 :[!1] ¡! ! de¯ned recursively by a ½0(®; ¯) = hjC¯ \ ®ji ½0(®; min(C¯ n ®)); 2 a with the boundary value ½0(®; ®) = ; where the symbol refers to the sequence obtained by concatenating the one term sequence hjC¯ \ ®ji with the already known ¯nite sequence ½0(®; min(C¯ n ®) of integers. Knowing ½0(®; ¯) and the ordinal ¯ one can reconstruct two following two traces of the walk from ¯ to ®. 1.4 De¯nition. The upper trace of the walk from ¯ to ® is the set Tr(®; ¯) = f» : ½0(»; ¯) v ½0(®; ¯)g; where v denotes the ordering of being an initial segment among ¯nite se- quences of ordinals. One also has the recursive de¯nition of this trace, Tr(®; ¯) = f¯g [ Tr(®; min(C¯ n ®)) with the boundary value Tr(®; ®) = f®g: 1.5 De¯nition. The lower trace of the walk from ¯ to ® is the set L(®; ¯) = f¸»(®; ¯): ½0(»; ¯) < ½0(®; ¯)g; where < is the ordering of being a strict initial segment among the ¯nite se- quences of ordinals and where for an ordinal » such that ½0(»; ¯) < ½0(®; ¯); ¸»(®; ¯) = maxfmax(C´ \ ®): ½0(´; ¯) v ½0(»; ¯)g: Note the following recursive de¯nition of this trace, L(®; ¯) = L(®; min(C¯ n ®)) [ (fmax(C¯ \ ®)g n max(L(®; min(C¯ n ®)))) with the boundary value L(®; ®) = ;: The following immediate facts about these notions will be frequently and very often implicitly used. 2 2 Note that while ½0 operates on the set [!1] of unordered pairs of countable ordinals, the formal recursive de¯nition of ½0(®; ¯) does involves the term ½0(®; ®) and so we have to give it as a boundary value ½0(®; ®) = ;. The boundary value is always speci¯ed when such a function is recursively de¯ned and will typically be either 0 or the empty sequence 2 ;. This convention will be used even when a function e with domain such as [!1] is given to us beforehand and we consider its diagonal value e(®; ®): 1. Walks in the space of countable ordinals 5 1.6 Lemma. For ® · ¯ · γ; 3 a (a) ® > L(¯; γ) implies that ½0(®; γ) = ½0(¯; γ) ½0(®; ¯) and therefore that Tr(®; γ) = Tr(¯; γ) [ Tr(®; ¯); (b) L(®; ¯) > L(¯; γ) implies that L(®; γ) = L(¯; γ) [ L(®; ¯): a 1.7 De¯nition. The full lower trace of the minimal walk is the function 2 <! F :[!1] ¡! [!1] de¯ned recursively by [ F (®; ¯) = F (®; min(C¯ n ®)) [ F (»; ®); »2C¯ \® with the boundary value F (®; ®) = f®g for all ®.
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