Linear ROD subsets of Borel partial orders are countably cofinal in the Solovay model
aa r X i v : . [ m a t h . L O ] J un Linear ROD subsets of Borel partial orders arecountably cofinal in Solovay’s model
Vladimir Kanovei ∗ May 16, 2017
Abstract
The following is true in the Solovay model.1. If h D ; ≤i is a Borel partial order on a set D of the reals, X ⊆ D is a ROD set, and ≤ ↾ X is linear, then ≤ ↾ X is countably cofinal.2. If in addition every countable set Y ⊆ D has a strict upperbound in h D ; ≤i then the ordering h D ; ≤i has no maximal chainsthat are ROD sets. Linear orders, which typically appear in conventional mathematics, arecountably cofinal. In fact any
Borel (as a set of pairs) linear order on asubset of a Polish space is countably cofinal: see, e. g. , [1]. On the otherhand, there is an uncountably-cofinal quasi-order of class Σ on NN . Example 1.
Fix any recursive enumeration Q = { q k : k ∈ N } of the ra- ←− ex tionals. For any ordinal ξ < ω , let X ξ be the set of all points x ∈ NN such that the maximal well-ordered (in the sense of the usual order of therationals) initial segment of the set Q x = { q k : x ( k ) = 0 } has the order type ξ . Thus NN = S ξ<ω X ξ . For x, y ∈ NN define x y iff x ∈ X ξ , y ∈ X η ,and ξ ≤ η . Thus is a prewellordering of length exactly ω . It is a routineexercise to check that belongs to Σ .We can even slightly change the definition of to obtain a true linearorder. Define x ′ y iff either x ∈ X ξ , y ∈ X η , and ξ < η , or x, y ∈ X ξ for one and the same ξ and x < y in the sense of the lexicographical linearorder on NN . Clearly ′ is a linear order of cofinality ω and class Σ .Yet there is a rather representative class of ROD (that is, real-ordinaldefinable) linear orderings which are consistently countably cofinal. This isthe subject of the next theorem. ∗ IPPI, Moscow, Russia. heorem 2. The following sentence is true in the Solovay model : if ≤ is ←− m a Borel partial quasi-order on a (Borel) set D ⊆ NN , X ⊆ D is a ROD set,and ≤ ↾ X is a linear quasi-order, then ≤ ↾ X is countably cofinal. A partial quasi-order , PQO for brevity, is a binary relation ≤ satisfying x ≤ y ∧ y ≤ z = ⇒ x ≤ z and x ≤ x on its domain. In this case, an associatedequivalence relation ≡ and an associated strict partial order < are definedso that x ≡ y iff x ≤ y ∧ y ≤ x , and x < y iff x ≤ y ∧ y x . A PQO is linear , LQO for brevity, if we have x ≤ y ∨ y ≤ x for all x, y in its domain.A PQO h X ; ≤i (meaning: X is the domain of ≤ ) is Borel iff the set X is a Borel set in a suitable Polish space X , and the relation ≤ (as a set ofpairs) is a Borel subset of X × X .Thus it is consistent with ZFC that
ROD linear suborders of BorelPQOs are necessarily countably cofinal. Accordingly it is consistent with ZF + DC that any linear suborders of Borel PQOs are countably cofinal.By the Solovay model we understand a model of ZFC in which all
ROD sets of reals have some basic regularity properties, for instance, are Lebesguemeasurable, have the Baire property, see [6]. We’ll make use of the followingtwo results related to the Solovay model.
Proposition 3 (Stern [7]) . It holds in the Solovay model that if ρ < ω ←− p2 then there is no ROD ω -sequence of pairwise different sets in Σ ρ . Proposition 4.
It holds in the Solovay model that if ≤ is a ROD
LQO ←− p3 on a set D ⊆ NN then there exist a ROD antichain A ⊆ <ω and a ROD map ϑ : D −→ A such that x ≤ y ⇐⇒ ϑ ( x ) ≤ lex ϑ ( y ) for all x, y ∈ D . A few words on the notation. The set 2 <ω = S ξ<ω ξ consists of alltransfinite binary sequences of length < ω , and if ξ < ω then 2 ξ is the setof all binary sequences of length exactly ξ . A set A ⊆ <ω is an antichain if we have s t for any s, t ∈ A , where s ⊂ t means that t is a properextension of s . By ≤ lex we denote the lexicographical order on 2 <ω , thatis, if s, t ∈ <ω then s ≤ lex t iff either 1) s = t or 2) s t , t s , andthe least ordinal ξ < dom s , dom t such that s ( ξ ) = t ( ξ ) satisfies s ( ξ ) < t ( ξ ).Obviously ≤ lex linearly orders any antichain A ⊆ <ω .Proposition 4 follows from Theorem 6 in [5] saying that if, in the Solovaymodel, ≤ is a ROD
PQO on a set D ⊆ NN then: either a condition (I s ) holds, which for LQO relations ≤ is equivalent tothe existence of A and ϑ as in Proposition 4, or a condition (II) holds, which is incompatible with ≤ being a LQO.2hus we obtain Proposition 4 as an immediate corollary.The next simple fact will be used below. Lemma 5. If ξ < ω then any set C ⊆ ξ is countably ≤ lex -cofinal, that ←− cc is, there is a set C ′ ⊆ C , at most countable and ≤ lex -cofinal in C . Proof (Theorem 2) . We argue in the Solovay model.
Suppose that ≤ is aBorel PQO on a (Borel) set D ⊆ NN , X ⊆ D is a ROD set, and ≤ ↾ X isa LQO. Our goal will be to show that ≤ ↾ X is countably cofinal, that is,there is a set Y ⊆ X , at most countable and ≤ -cofinal in X .The restricted order ≤ ↾ X is ROD , of course, and hence, by Proposition 4,there is a
ROD map ϑ : X −→ A onto an antichain A ⊆ <ω (also obviouslya ROD set) such that x ≤ y ⇐⇒ ϑ ( x ) ≤ lex ϑ ( y ) for all x, y ∈ X .If ξ < ω then let A ξ = A ∩ ξ and X ξ = { x ∈ D : ϑ ( x ) ∈ A ξ } . Case 1 : there is an ordinal ξ < ω such that A ξ is ≤ lex -cofinal in A .However, by Lemma 5, there is a set A ′ ⊆ A ξ , at most countable and ≤ lex -cofinal in A ξ , and hence ≤ lex -cofinal in A as well by the choice of ξ . If s ∈ A ′ then pick an element x s ∈ X such that ϑ ( x s ) = s . Then the set Y = { x s : s ∈ A ′ } is a countable subset of X, ≤ -cofinal in X, as required. Case 2 : not Case 1. That is, for any η < ω there is an ordinal ξ < ω and an element s ∈ A ξ such that η < ξ and t < lex s for all t ∈ A η . Thenthe sequence of sets D ξ = { z ∈ D : ∃ x ∈ X ( z ≤ x ∧ ϑ ( x ) ∈ A ξ ) } is ROD and has uncountably many pairwise different terms.We are going to get a contradiction. Recall that ≤ is a Borel relation,hence it belongs to Σ ρ for an ordinal 1 ≤ ρ < ω . Now the goal is to provethat all sets D ξ belong to Σ ρ as well — this contradicts to Proposition 3,and the contradiction accomplishes the proof of the theorem.Consider an arbitrary ordinal ξ < ω . By Lemma 5 there exists a count-able set A ′ = { s n : n < ω } ⊆ A ξ , ≤ lex -cofinal in A ξ . If n < ω then pickan element x n ∈ X such that ϑ ( x n ) = s n . Note that by the choice of ϑ any other element x ∈ X with ϑ ( x ) = s n satisfies x ≡ x n , where ≡ is theequivalence relation on D associated with ≤ . It follows that D ξ = S n X n , where X n = { z ∈ D : z ≤ x n } , so each X n is a Σ ρ set together with ≤ , and so is D ξ as a countable unionof sets in Σ ρ . ( Theorem 2 )3e continue with a few remarks and questions.
Problem 6.
Can one strengthen Theorem 2 as follows: the restricted re-lation ≤ ↾ X has no monotone ω -sequences ? Lemma 5 admits such astrengthening: if ξ < ω then easily any ≤ lex -monotone sequence in 2 ξ iscountable.Using Shoenfield’s absoluteness, we obtain: Corollary 7. If ≤ is a Borel PQO on a (Borel) set D ⊆ NN , X ⊆ D is a ←− mc Σ set, and ≤ ↾ X is a linear quasi-order, then ≤ ↾ X is countably cofinal. Note that Corollary 7 fails for arbitrary LQOs of class Σ (that is, notnecessarily linear suborders of Borel PQOs), see Example 1. Proof.
In the case considered, the property of countable cofinality of ≤ ↾ X can be expressed by a Σ formula. Thus it remains to consider a Solovay-type extension of the universe and refer to Theorem 2. Yet there is a really elementary proof of Corollary 7.Let Y be the set of all elements y ∈ D ≤ -comparable with every element x ∈ X . This is a Σ set, and X ⊆ Y (as ≤ is linear on X ). Thereforethere is a Borel set Z such that X ⊆ Z ⊆ Y . Now let U be the set of all z ∈ Z ≤ -comparable with every element y ∈ Y . Still this is a Σ set, and X ⊆ U by the definition of Y . Therefore there is a Borel set W such that X ⊆ W ⊆ U . And by definition still ≤ is linear on W . It follows that W does not have increasing ω -sequences, and hence neither does X . Problem 8.
Is Corollary 7 true for Π sets X ?We cannot go much higher though. Indeed, if ≤ is, say, the eventualdomination order on NN , then the axiom of constructibility implies the ex-istence of a ≤ -monotone ω -sequence of class ∆ .Now a few words on Borel PQOs ≤ having the following property:( ∗ ) if X is a countable set in the domain of ≤ then there is an element y such that x < y (in the sense of the corresponding strict ordering) forall x ∈ X .A thoroughful study of some orderings of this type (for instance, the orderingon R ω defined so that x ≤ y iff either x ( n ) = y ( n ) for all but finite n or x ( n ) < y ( n ) for all but finite n ) was undertaken in early papers of Felix We’ll not discuss the issue of an inaccessible cardinal on the background. e. g. , [2, 3] (translated to English in [4]). In particular, Hausdorffinvestigated the structure of pantachies , that is, maximal linearly orderedsubsets of those partial orderings. As one of the first explicit applications ofthe axiom of choice, Hausdorff established the existence of a pantachy in anypartial order, and made clear distinction between such an existence proofand an actual, well-defined construction of an individual pantachy (see [2],p. 110). The next result shows that the latter is hardly possible in
ZFC ,at least if we take for granted that any individual set-theoretic constructionresults in a
ROD set.
Corollary 9.
The following sentence is true in the Solovay model : if ≤ is ←− l a Borel partial quasi-order on a (Borel) set D ⊆ NN , satisfying ( ∗ ) , then ≤ has no ROD pantachies.
Proof.
It follows from ( ∗ ) that any pantachy in h D ; ≤i is a set of uncount-able cofinality. Now apply Theorem 2.A further corollary: it is impossible to prove the existence of pantachiesin any Borel PQO satisfying ( ∗ ) in ZF + DC . References [1] L. A. Harrington, D. Marker, and S. Shelah, Borel orderings.
Trans. Amer.Math. Soc. , 1988, 310, pp. 293–302.[2] F. Hausdorff, Untersuchungen ¨uber Ordnungstypen IV, V.
Ber. ¨uber dieVerhandlungen der K¨oniglich S¨achsische Gesellschaft der Wissenschaften zuLeipzig, Math.-phys. Klasse , 1907, 59, pp. 84–159.[3] F. Hausdorff, Die Graduirung nach dem Endverlauf.
Abhandlungen derK¨oniglich S¨achsische Gesellschaft der Wissenschaften zu Leipzig, Math.-phys.Klasse , 1909, 31, pp. 295–334.[4] F. Hausdorff,
Hausdorff on ordered sets , Translated from the German, editedand with commentary by J. M. Plotkin. History of Mathematics, 25. AMS,Providence, RI and LMS, London, 2005. xviii+322 pp.[5] Vladimir Kanovei. Linearization of definable order relations.
Annals of Pureand Applied Logic , 2000, 102, 1-2, pp. 69–100.[6] R.M. Solovay. A model of set-theory in which every set of reals is Lebesguemeasurable.
Ann. Math. (2) , 1970, 92, pp. 1–56.[7] J. Stern. On Lusin’s restricted continuum problem.
Ann. Math. (2) , 1984,120, pp. 7–37., 1984,120, pp. 7–37.