Descriptive Set Theory and Forcing: How to Prove Theorems about Borel Sets the Hard Way 9781316716977

Table of contents :
Contents......Page 7
1 What are the reals, anyway?......Page 10
2 Borel Hierarchy......Page 12
3 Abstract Borel hierarchies......Page 16
4 Characteristic function of a sequence......Page 18
5 Martin's Axiom......Page 21
6 Generic Gδ......Page 23
7 α-forcing......Page 26
8 Boolean algebras......Page 31
9 Borel order of a field of sets......Page 35
10 CH and orders of separable metric spaces......Page 37
11 Martin-Solovay Theorem......Page 39
12 Boolean algebra of order ω1......Page 43
13 Luzin sets......Page 47
14 Cohen real model......Page 51
15 The random real model......Page 62
16 Covering number of an ideal......Page 69
17 Analytic sets......Page 73
18 Constructible well-orderings......Page 76
19 Hereditarily countable sets......Page 77
20 Shoenfield Absoluteness......Page 79
21 Mansfield-Solovay Theorem......Page 81
22 Uniformity and Scales......Page 82
23 Martin's axiom and Constructibility......Page 87
24 Σ 1 2 well-orderings......Page 89
25 Large Π 1 2 sets......Page 90
26 Souslin-Luzin Separation Theorem......Page 93
27 Kleene Separation Theorem......Page 95
28 Π 1 1-Reduction......Page 98
29 Δi-codes......Page 100
30 Π 1 1 equivalence relations......Page 103
31 Borel metric spaces and lines in the plane......Page 108
32 Σ 1 1 equivalence relations......Page 112
33 Louveau's Theorem......Page 116
34 Proof of Louveau's Theorem......Page 122
References......Page 126
Index......Page 133
Elephant Sandwiches......Page 135

Citation preview

Descriptive Set Theory and Forcing Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. In this volume, the 4th publication in the Lecture Notes in Logic series, Miller develops the necessary features of the theory of descriptive sets in order to present a new proof of Louveau’s separation theorem for analytic sets. While some background in mathematical logic and set theory is assumed, the material is based on a graduate course given by the author at the University of Wisconsin, and is thus accessible to students and researchers alike in these areas, as well as in mathematical analysis. A r n o l d W. M i l l e r works in the Department of Mathematics at the University of Wisconsin, Madison.

L E C T U R E N OT E S I N L O G I C

A Publication of The Association for Symbolic Logic This series serves researchers, teachers, and students in the field of symbolic logic, broadly interpreted. The aim of the series is to bring publications to the logic community with the least possible delay and to provide rapid dissemination of the latest research. Scientific quality is the overriding criterion by which submissions are evaluated. Editorial Board Jeremy Avigad, Department of Philosophy, Carnegie Mellon University Zoe Chatzidakis DMA, Ecole Normale Supérieure, Paris Peter Cholak, Managing Editor Department of Mathematics, University of Notre Dame, Indiana Volker Halbach, New College, University of Oxford H. Dugald Macpherson School of Mathematics, University of Leeds Slawomir Solecki Department of Mathematics, University of Illinois at Urbana–Champaign Thomas Wilke, Institut für Informatik, Christian-Albrechts-Universität zu Kiel More information, including a list of the books in the series, can be found at http://www.aslonline.org/books-lnl.html

L E C T U R E N OT E S I N L O G I C 4

Descriptive Set Theory and Forcing How to Prove Theorems about Borel Sets the Hard Way ARNOLD W. MILLER University of Wisconsin, Madison

association for symbolic logic

University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 4843/24, 2nd Floor, Ansari Road, Daryaganj, Delhi – 110002, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107168060 10.1017/9781316716977 First edition © 1995 Springer-Verlag Berlin Heidelberg This edition © 2016 Association for Symbolic Logic under license to Cambridge University Press. Association for Symbolic Logic Richard A. Shore, Publisher Department of Mathematics, Cornell University, Ithaca, NY 14853 http://www.aslonline.org This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. A catalogue record for this publication is available from the British Library. ISBN 978-1-107-16806-0 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate.

Note to the readers Departing from the usual author's statement-I would like to say that I am not responsible for any of the mistakes in this document. Any mistakes here are the responsibility of the reader. If anybody wants to point out a mistake to me, I promise to respond by saying "but you know what I meant to say, don't you?" These are lecture notes from a course I gave at the University of Wisconsin during the Spring semester of 1993. Some knowledge of forcing is assumed as well as a modicum of elementary Mathematical Logic, for example, the LowenheimSkolem Theorem. The students in my class had a one semester course, introduction to mathematical logic covering the completeness theorem and incompleteness theorem, a set theory course using Kunen [54], and a model theory course using Chang and Keisler [17]. Another good reference for set theory is Jech [43]. Oxtoby [88] is a good reference for the basic material concerning measure and category on the real line. Kuratowski [57] and Kuratowski and Mostowski [58] are excellent references for classical descriptive set theory. Moschovakis [87] and Kechris [52] are more modern treatments of descriptive set theory. The first part is devoted to the general area of Borel hierarchies, a subject which has always interested me. The results in section 14 and 15 are new and answer questions from my thesis. I have also included (without permission) an unpublished result of Fremlin (Theorem 13.4). Part II is devoted to results concerning the low projective hierarchy. It ends with a theorem of Harrington from his thesis that is consistent to have IJ2 sets of arbitrary size. The general aim of part III and IV is to get to Louveau's theorem. Along the way many of the classical theorems of descriptive set theory are presented "just-in-time" for when they are needed. This technology allows the reader to keep from overfilling his or her memory storage device. I think the proof given of Louveau's Theorem 33.1 is also a little different. * Questions like "Who proved what?" always interest mς, so I have included my best guess here. Hopefully, I have managed to offend a large number of mathematicians.

1 In a randomly infinite Universe, any event occurring here and now with finite probability must be occurring simultaneously at an infinite number of other sites in the Universe. It is hard to evaluate this idea any further, but one thing is certain: if it is true then it is certainly not original!- The Anthropic Cosmological Principle, by John Barrow and Frank Tipler.

Contents 1 What are the reals, anyway?

5

1

7

On the length of Borel hierarchies

2 Borel Hierarchy

7

3 Abstract Borel hierarchies

11

4 Characteristic function of a sequence

13

5 Martin's Axiom

16

6 Generic G&

18

7 α-forcing

21

8 Boolean algebras

26

9 Borel order of a field of sets

30

10 CH and orders of separable metric spaces

32

11 Martin-Solovay Theorem

34

12 Boolean algebra of order ω\

38

13 Luzin sets

42

14 Cohen real model

46

15 The random real model

57

16 Covering number of an ideal

64

II

68

Analytic sets

17 Analytic sets

68

18 Constructible well-orderings

71

19 Hereditarily countable sets

72

20 Shoenfield Absoluteness

74

21 Mansfield-Solovay Theorem

76

22 Uniformity and Scales

78

23 Martin's axiom and Constructibility

82

24 Σ2 well-orderings

84

25 Large Π| sets

85

III

88

Classical Separation Theorems

26 Souslin-Luzin Separation Theorem

88

27 Kleene Separation Theorem

90

28 Πj-Reduction

93

29 Δi-codes

95

IV

98

Gandy Forcing

30 U{ equivalence relations

98

31 Borel metric spaces and lines in the plane

103

32 Σ} equivalence relations

107

33 Louveau's Theorem

111

34 Proof of Louveau's Theorem

117

References Index Elephant Sandwiches

121 128 130

1

What are the reals, anyway?

Definitions. Let ω = {0,1,...} and let ωω (Baire space) be the set of functions fromωtou;. Let ω 2 and r(s~n) increases to λ as n —• 0 0 .

22

7 a-FORCING

It is easy to see that for every a < ω\ nice α-trees exist. For X a Hausdorff space with countable base, B, and T a nice α-tree (a > 2), define the partial order JP = F(X, B, T) which we call a-forctng as follows: p£ψif[p=

(t,F) where

1. t : D->B where DCΓ° = {sGT: φ ) = 0} is finite, 2. F C Γ

> o

x I is finite where T>0 = T \ T° = {s G T : φ ) > 0},

3. if (s,x),(s Λ n,ί/) G F , then x φ y> and 4. if (s,z) G F a n d ^ ( s Λ n ) = : 5 , then x £ B. The ordering on F is given by p < q iff tp D tq and Fp D Fq. Lemma 7.2 P Λαs ccc. proof: Suppose A is uncountable antichain. Since there are only countably many different tp without loss we may assume that there exists t such that tp = t for all p G A. Consequently for p,q G A the only thing that can keep pU q from being a condition is that there must be an x G X and an s, s~n G T>0 such that

But now for each p e A let Hp : X -^ [T>0] 0 (5,x) G F p } . Then {/fp : p G A} is an uncountable antichain in the order of finite partial functions from X to [ T t > 0 ] < ω , a countable set.

•

Define for G a F-filter the set Us C X for s G T as follows: 1. for s G T° let {/, = J5 iff Ξp G G such that tp(s) = JB and 2. for * € Γ>° let tf, = Π n e w ~ t V n Note that t/5 is a Π$(X)-set where r(s) = β. Lemma 7.3 IfG is Ψ-generic over V then in V[G] we have that for every x G X and s G T>0 xeUs 3peG (5, x) G Fp.

23

Descriptive Set Theory and Forcing

proof: First suppose that r(s) — 1 and note that the following set is dense: D = {peΨ:(s,x)eFp

or 3n3B G B x G B and tp(s~n) = B}.

To see this let p G P be arbitrary. If (s, x) G Fp then p e D and we are already done. If (s, x) £ Fp then let Y = {y

(s,y)eFp}.

Choose B G B with x E B and y disjoint from B. Choose s~n not in the domain of tpj and let q = (tqiFp) be defined by ^ = tp U (s~n, 5 ) . So q < p and q e D. Hence D is dense. Now by definition x G Us iff x G Πn€u> ~ ^ V n So let G be a generic filter and p E G Π D. If (s, x) G Fp then we know that for every q G G and for every n, if tq(s~n) — B then x £ B. Consequently, x G Us. On the other hand if tp(s"n) = B where x G 5 , then x £ Us and for every g G G it must be that (s, 3?) ^ F ? (since otherwise p and g would be incompatible). Now suppose r(s) > 1. In this case note that the following set is dense: Λ

E = {p G P : (β, a?) G Fp or Ξn (s n, x) G F p } . To see this let p G P be arbitrary. Then either (s, x) G .Fp and already p € E oτ by choosing n large enough q = (ί p , F p U {(s~n, x)}) G ί?. (Note r(s~n) > 0.) Now assume the result is true for all Us~n. Let p EGΠE. If (5, x) G F p then for every q G G and n we have (s Λ n, x) ^ JF^ and so by induction x ζέ ί/βΛn and so x £ Us. On the other hand if (s"ny x) E Fp, then by induction x £Us-n and so x ^ ί/s, and so again for every q G G we have (5, x) ^ i ^ .

•

The following lemma is the heart of the old switcheroo argument used in Theorem 6.2. Given any Q C X define the rank(p, Q) as follows: rank(p, Q) = max{r(s) : (s, x) G Fp for some x G X \ Q). L e m m a 7.4 (Rank Lemma). For any β > 1 and p G P /Λere ezisfc p compatible with p such that 1. rank(p, Q) < β + 1 αnrf 2. for any qeΨ i/rank(g,Q) < β, then p and q compatible implies p and q compatible. proof: Let po < p be any extension which satisfies: for any (s, x) G Fp and n G w, if r(s) = λ > β is a limit ordinal and r(s"n) < β + 1, then there exist m G w such that (s"n~m, x) G Fpo Note that since r(s~n) is increasing to λ there are

24

7 a-FORCING Λ

only finitely many (s,x) and s n t o worry about. Also r(s"n~m) > 0 so this is possible to do. Now let p be defined as follows: tp = tp and i ^ = {(s, x)eFPo

:xeQ

oτ r(s) < β + 1}.

Suppose for contradiction that there exists q such that rank(g,Q) < /?, p and g compatible, but p and q incompatible. Since p and q are incompatible either 1. there exists (s, x) G Fq and tp(s"n) = B with x G £, or 2. there exists (s, a?) G F p and tq(s~n) = £? with x G 5, or 3. there exists (s,x) G F p and (s"n,x) G F g , or 4. there exists (s,x) G F g and (s Λ n,x) G F p . (1) cannot happen since tp = *p and so p, g would be incompatible. (2) cannot happen since r(s) = 1 and β > 1 means that (s, a?) G Fp and so again p and g are incompatible. If (3) or (4) happens for x G Q then again (in case 3) (s, x) G Fp or (in case 4) (s~n, x) G Fp and so p, g incompatible. So assume x £ Q. In case (3) by the definition of rank(g, Q) < β we know that r(s"n) < β. Now since T is a nice tree we know that either r(s) < β and so (5, x) G Fp or r(s) = λ a limit ordinal. Now if λ < β then (s, a?) G Fp. If λ > β then by our construction of po there exist m with (s Λ n Λ m, x) G Fp and so p, g are incompatible. Finally in case (4) since x £ Q and so r(s) < β we have that r(s"n) < β and so (s~n, x) G Fp and so p, g are incompatible.

•

Intuitively, it should be that statements of small rank are forced by conditions of small rank. The next lemma will make this more precise. Let L^ (Pa : a < K) be the infinitary propositional logic with {P α : α < «} as the atomic sentences. Let Πo-sentences be the atomic ones, {Pa : a < K}. For any β > 0 let 0 be a Ilβ-sentence iff there exists Γ C I L . Λ IL-sentences and

Models for this propositional language can naturally be regarded as subsets Y G K where we define 1. y

μPαiffαey,

2. Y \= -1(9 iff not y |=0, and 3. y |= M Γ iffy (= 0 for every 0 G Γ.

25

Descriptive Set Theory and Forcing

L e m m a 7.5 (Rank and Forcing Lemma) Suppose rank : F —• OR is any funcQ

tion on a poset F which satisfies the Rank Lemma 7.4- Suppose \\~χYC K and for every p £ F and a < K if p \\~ α EY then there exist p compatible with p such that rank(p) = 0 and P

\\- a ey .

Then for every Up-sentence θ (in the ground model) and every p £ F, if p\\-

"γ\=θ»

then there exists p compatible with p such that rank(p) < β and p\\-

"γ\=θ".

proof: This is one of those lemmas whose statement is longer than its proof. The proof is induction on β and for β = 0 the conclusion is true by assumption. So suppose β > 0 and θ = /)(\t/)GΓ ~^Φ where Γ C \Jδ ord is a rank function and E C IB is a countable collection of rank zero elements, then for any a G JJy(E) and aφQ there exists b < a with r(b) < 7. proof: To see this let E = {en generic extension

0

: n G ω} and let Y be a name for the set in the Y =

{neω:eneG}.

0

Note that e n = [ n £Y ]. For elements 6 of IB in the complete subalgebra generated by E let us associate sentences 0& of the infinitary propositional logic Loo(Pn : n G ω) as follows: θen = Pn

Note that [ Y |= Θb ] = 6 and if 6 G U^(E) then θb is a Π 7 -sentence. The Rank and Forcing Lemma 7.5 gives us (by translating p \\- Y \=z θb into p < [ Y \= θb j = 6) that: For any 7 > 1 and p < b G JJ®(E) there exists a p compatible with p such that p < 6 and r(p) < 7.

• Now we use the lemmas to see that ord(IB) > α. Given any countable E C IB, let Q C X be countable so that for any e G £ there exists if C IP countable so that e = Σ H and for every p E H we have rank(p,Q) = 0. Let x e X\Q be arbitrary; then we claim:

Descriptive Set Theory and Forcing

29

We have chosen Q so that r(p) = rank(p, Q) = 0 for any p G E so the hypothesis of Lemma 8.4 is satisfied. Suppose for contradiction that [ x E U{) 1 = b e Σ° ( £ ) . Let b = Σn€ω bn where each bn is Π° n (C) for some Ίn < α. For some n and p G F w e would have p < bn. By Lemma 8.4 we have that there exists p with p < bn < b = [ x G UQ J and rank(p, Q) < yn. But by the definition of rank(p,Q) the pair ((),#) is not in Fp, but this contradicts

p•

This takes care of all countable successor ordinals. (We leave the case of α = 0,1 for the reader to contemplate.) For λ a limit ordinal take ctn increasing to λ and let F = Σn 2, and X is super-I-Luzin. Then oτd(X) = a. proof: Note that the oτά(X) is the minimum a such that for every B G Borel(2α;) there exists A € Π°(2 ω ) with AΠX = BOX. Since ord(C) = a we know that given any Borel set B there exists a Π° set A such that AAB G /. Since X is Luzin we know that X Π (AΔJB) is countable. Hence there exist countable sets Fo,F\ such that

XΓιB = Xn((A\F0)UF1). But since a > 2 we have that ((A \ Fo) U Fi) is also Π° and hence ord(X) < α. On the other hand for any β < a we know there exists a Borel set B such that for every Jλφ set A we have BAA £ I (since ord(C) > β). But since X is super-/-Luzin we have that for every Πjg set A that X Π (BAA) φ 0 and hence

XΠBφ

XΓ)A. Consequently, oτd(X) > β.

m Corollary 10.4 (CH) For every a < ω\ there exists a separable metric space X such that ord(X) = a. While a graduate student at Berkeley I had obtained the result that it was consistent with any cardinal arithmetic to assume that for every a < ω\ there exists a separable metric space X such that ord(X) = α. It never occurred to me at the time to ask what CH implied. In fact, my way of thinking at the time was that proving something from CH is practically the same as just showing it is consistent. I found out in the real world (outside of Berkeley) that they are considered very differently. In Miller [73] it is shown that for every a < ω\ it is consistent there exists a separable metric space of order β iff a < β < ω\. But the general question is open. Question 10.5 For what C C ω\ is it consistent that C = {oτd(X) : X separable metric }?

34

11 MARTIN-SOLOVAY THEOREM

11

Martin-Solovay Theorem

In this section we the theorem below. The technique of proof will be used in the next section to produce a boolean algebra of order ω\. Theorem 11.1 (Martin-Solovay [72]) The following are equivalent for an infinite cardinal K: 1. MA K ; i.e., for any poset F which is ccc and family V of dense subsets ofψ with \V\< K there exists a Ψ-filterG with GO D ψ 0 for all D eV 2. For any ccc σ-ideal I in Borel(2ω) and X C / with \1\ < K we have that

ω

L e m m a 11.2 Let 1 = Borel(2 )/7 for some ccc σ-ideal I and let P = 1 \ {0}. The following are equivalent for an infinite cardinal K: 1. for any family V of dense subsets ofψ with \V\ < K, there exists a Ψ-filter G withGΠDφHiforallDeV 2. for any family T C Iff*' with \T\< K there exists an ultrafilterU on M which is T-complete, i.e., for every (bn : n G ω) £ T ιff3nbn€U

eU

n

3. for anyl

Cl with \1\ < K

proof: To see that (1) implies (2) note that for any (6n : n G ω) G W the set D = {p G Ψ : p < - Y^bn or 3n p < bn} n

is dense. Note also that any filter extends to an ultrafilter. To see that (2) implies (3) do as follows. Let H 7 stand for the family of sets whose transitive closure has cardinality less than the regular cardinal 7, i.e. they are hereditarily of cardinality less than 7. The set H 7 is a natural model of all the axioms of set theory except possibly the power set axiom, see Kunen [54]. Let M be an elementary substructure of H 7 for sufficiently large 7 with

|M| ω by p £ G iff p C x. This map has the property that for every s £ T > 0 the value of x(s) is the unique element of ω not in {x(s~n) : n £ ω}. proof: For each s £ T the set Ds = {p : s £ domain(p)} is dense. Also for each s £ T > 0 and k £ ω the set Eg={p:

p(s) = k or 3n p(s~n) = k}

is dense.

•

The poset ΨQ is separative, since if p jt q then either p and q are incompatible or there exists s £ domain(ρ) \ domain(p) in which case we can find p < p with p(s) φ q(s). Now if P α C IB is dense in the cBa I, it follows that for each p £ Ψa p=[pCx]

and for any s £ T>0 and k

[*(«) = * ] = Consequently if C = {p £ Pα : domain(p) C Γ 0 } then C C IB has the property that ord(C) = a + 1.

40

12 BOOLEAN ALGEBRA OF ORDER

Now let Σa