Regularity and Definability
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1 Regularity and Definability Yurii Khomskii University of Amsterdam PhDs in Logic III Brussels, 17 February 2011 Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 1 / 21
2 The setting The setting The continuum (R, R 2, P(ω), ω ω, 2 ω,... ). Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 2 / 21
3 The setting The setting The continuum (R, R 2, P(ω), ω ω, 2 ω,... ). Subsets of the continuum objects in space. y x Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 2 / 21
4 The setting The setting The continuum (R, R 2, P(ω), ω ω, 2 ω,... ). Subsets of the continuum objects in space. Regularity properties vs. definability of these objects. y x Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 2 / 21
5 The setting The setting The continuum (R, R 2, P(ω), ω ω, 2 ω,... ). Subsets of the continuum objects in space. Regularity properties vs. definability of these objects. 1 Regularity. Lebesgue measure, Baire property, Ramsey property,... y x Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 2 / 21
6 The setting The setting The continuum (R, R 2, P(ω), ω ω, 2 ω,... ). Subsets of the continuum objects in space. Regularity properties vs. definability of these objects. 1 Regularity. Lebesgue measure, Baire property, Ramsey property,... 2 Definability. y Classifying sets according to logical complexity. x Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 2 / 21
7 The setting The setting The continuum (R, R 2, P(ω), ω ω, 2 ω,... ). Subsets of the continuum objects in space. Regularity properties vs. definability of these objects. 1 Regularity. Lebesgue measure, Baire property, Ramsey property,... 2 Definability. y Classifying sets according to logical complexity. 3 Relationship between these. x Independence from ZFC (forcing extensions over L). Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 2 / 21
8 Regularity 1. Regularity Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 3 / 21
9 Regularity Regularity What do we mean by regularity properties of sets of reals? Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 4 / 21
10 Regularity Regularity What do we mean by regularity properties of sets of reals? Example 1. Lebesgue measure. Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 4 / 21
11 Regularity Regularity What do we mean by regularity properties of sets of reals? Example 1. Lebesgue measure. For q < q Q, µ([q, q ]) := q q. Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 4 / 21
12 Regularity Regularity What do we mean by regularity properties of sets of reals? Example 1. Lebesgue measure. For q < q Q, µ([q, q ]) := q q. Naturally extend to Borel subsets of R. Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 4 / 21
13 Regularity Regularity What do we mean by regularity properties of sets of reals? Example 1. Lebesgue measure. For q < q Q, µ([q, q ]) := q q. Naturally extend to Borel subsets of R. A R is Lebesgue-null if B Borel with A B and µ(b) = 0. Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 4 / 21
14 Regularity Regularity What do we mean by regularity properties of sets of reals? Example 1. Lebesgue measure. For q < q Q, µ([q, q ]) := q q. Naturally extend to Borel subsets of R. A R is Lebesgue-null if B Borel with A B and µ(b) = 0. A is Lebesgue-measurable if B Borel such that (A \ B) (B \ A) is Lebesgue-null. Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 4 / 21
15 Regularity Regularity What do we mean by regularity properties of sets of reals? Example 1. Lebesgue measure. For q < q Q, µ([q, q ]) := q q. Naturally extend to Borel subsets of R. A R is Lebesgue-null if B Borel with A B and µ(b) = 0. A is Lebesgue-measurable if B Borel such that (A \ B) (B \ A) is Lebesgue-null. Captures the intuition of size or volume of a set of reals ( object in space ). Can naturally be extended to R n. Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 4 / 21
16 Regularity Measure However, there are non-lebesgue-measurable sets (Vitali, 1905). Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 5 / 21
17 Regularity Measure However, there are non-lebesgue-measurable sets (Vitali, 1905). Proof. If A is Lebesgue-measurable then there exists a perfect set P with µ(p) > 0 s.t. P A or P A =. Use Axiom of Choice to diagonalize against perfect sets. Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 5 / 21
18 Regularity Measure However, there are non-lebesgue-measurable sets (Vitali, 1905). Proof. If A is Lebesgue-measurable then there exists a perfect set P with µ(p) > 0 s.t. P A or P A =. Use Axiom of Choice to diagonalize against perfect sets. Another proof. Let U be an ultrafiler on ω. Identify P(ω) with 2 ω, then U is non-lebesguemeasurable. Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 5 / 21
19 Regularity Measure However, there are non-lebesgue-measurable sets (Vitali, 1905). Proof. If A is Lebesgue-measurable then there exists a perfect set P with µ(p) > 0 s.t. P A or P A =. Use Axiom of Choice to diagonalize against perfect sets. Another proof. Let U be an ultrafiler on ω. Identify P(ω) with 2 ω, then U is non-lebesguemeasurable. Problematic consequences for spatial reasoning, e.g., Banach-Tarski paradox. Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 5 / 21
20 Regularity Other examples A R has the Baire property if B Borel such that (A \ B) (B \ A) is meager. Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 6 / 21
21 Regularity Other examples A R has the Baire property if B Borel such that (A \ B) (B \ A) is meager. A R is Marczewski-measurable if for every perfect set P there is a perfect subset Q P such that Q A or Q A =. Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 6 / 21
22 Regularity Other examples A R has the Baire property if B Borel such that (A \ B) (B \ A) is meager. A R is Marczewski-measurable if for every perfect set P there is a perfect subset Q P such that Q A or Q A =. Ramsey property, doughnut property, perfect set property, K σ -regularity,.... Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 6 / 21
23 Regularity Other examples A R has the Baire property if B Borel such that (A \ B) (B \ A) is meager. A R is Marczewski-measurable if for every perfect set P there is a perfect subset Q P such that Q A or Q A =. Ramsey property, doughnut property, perfect set property, K σ -regularity,.... In each case, we can find counterexamples. Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 6 / 21
24 Regularity Other examples A R has the Baire property if B Borel such that (A \ B) (B \ A) is meager. A R is Marczewski-measurable if for every perfect set P there is a perfect subset Q P such that Q A or Q A =. Ramsey property, doughnut property, perfect set property, K σ -regularity,.... In each case, we can find counterexamples. But... Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 6 / 21
25 Regularity Other examples A R has the Baire property if B Borel such that (A \ B) (B \ A) is meager. A R is Marczewski-measurable if for every perfect set P there is a perfect subset Q P such that Q A or Q A =. Ramsey property, doughnut property, perfect set property, K σ -regularity,.... In each case, we can find counterexamples. But... typical construction involves induction along a well-ordering of the continuum (Axiom of Choice). Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 6 / 21
26 Regularity Other examples A R has the Baire property if B Borel such that (A \ B) (B \ A) is meager. A R is Marczewski-measurable if for every perfect set P there is a perfect subset Q P such that Q A or Q A =. Ramsey property, doughnut property, perfect set property, K σ -regularity,.... In each case, we can find counterexamples. But... typical construction involves induction along a well-ordering of the continuum (Axiom of Choice). Question Can we find an explicit example of a non-regular set? (and what does that even mean?) Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 6 / 21
27 Definability 2. Definability Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 7 / 21
28 Definability Descriptive set theory Descriptive set theory: not just about sets, but about their descriptions or definitions. Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 8 / 21
29 Definability Descriptive set theory Descriptive set theory: not just about sets, but about their descriptions or definitions. Focus on second-order number theory (N 2 ): Variables range over natural numbers or real numbers. Natural number quantifiers: 0 0, Real number quantifiers: 1 1. Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 8 / 21
30 Definability Descriptive set theory Descriptive set theory: not just about sets, but about their descriptions or definitions. Focus on second-order number theory (N 2 ): Variables range over natural numbers or real numbers. Natural number quantifiers: 0 0, Real number quantifiers: 1 1. Complexity of N 2 -formulas: Σ 0 n, Π 0 n,..., Σ 1 n, Π 1 n,.... Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 8 / 21
31 Definability Complexity of sets Complexity of a set of reals measured by complexity of defining N 2 -formula. A = {x R N 2 = φ(x, a)} Note that we allow a fixed real parameter a R in the definition. Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 9 / 21
32 Definability Complexity of sets Complexity of a set of reals measured by complexity of defining N 2 -formula. A = {x R N 2 = φ(x, a)} Note that we allow a fixed real parameter a R in the definition. Definition We say A has complexity Σ i n (Π i n) iff φ has complexity Σ i n (Π i n). Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 9 / 21
33 Definability Complexity of sets Complexity of a set of reals measured by complexity of defining N 2 -formula. A = {x R N 2 = φ(x, a)} Note that we allow a fixed real parameter a R in the definition. Definition We say A has complexity Σ i n (Π i n) iff φ has complexity Σ i n (Π i n). Relation with topology: Σ 0 1 = open, Π 0 1 = closed, 1 1 = Borel, Σ 1 1 = analytic (continuous image of Borel). Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 9 / 21
34 Definability Hierarchy (open) (analytic) Σ 0 1 Σ 0 2 Σ 1 1 Σ 1 2 (Borel) Π 0 1 Π 0 2 Π 1 1 Π 1 2 (closed) (co-analytic) Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 10 / 21
35 Definability Hierarchy (open) (analytic) Σ 0 1 Σ 0 2 Σ 1 1 Σ 1 2 (Borel) Π 0 1 Π 0 2 Π 1 1 Π 1 2 (closed) (co-analytic) All Σ 1 1 sets are regular. Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 10 / 21
36 Definability Hierarchy (open) (analytic) Σ 0 1 Σ 0 2 Σ 1 1 Σ 1 2 (Borel) Π 0 1 Π 0 2 Π 1 1 Π 1 2 (closed) (co-analytic) All Σ 1 1 sets are regular. For many properties, also all Π 1 1 sets are regular. Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 10 / 21
37 Definability Hierarchy (open) (analytic) Σ 0 1 Σ 0 2 Σ 1 1 Σ 1 2 (Borel) Π 0 1 Π 0 2 Π 1 1 Π 1 2 (closed) (co-analytic) All Σ 1 1 sets are regular. For many properties, also all Π 1 1 sets are regular. Irregular sets (produced by AC) may lie far outside this hierarchy. Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 10 / 21
38 Definability Hierarchy (open) (analytic) Σ 0 1 Σ 0 2 Σ 1 1 Σ 1 2 (Borel) Π 0 1 Π 0 2 Π 1 1 Π 1 2 (closed) (co-analytic) All Σ 1 1 sets are regular. For many properties, also all Π 1 1 sets are regular. Irregular sets (produced by AC) may lie far outside this hierarchy. So paradoxes cannot occur if we restrict attention to analytic/co-analytic sets. Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 10 / 21
39 Definability Second level So on which level do things go wrong? Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 11 / 21
40 Definability Second level So on which level do things go wrong? Question: Does the assertion all Σ 1 2 sets are regular hold? Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 11 / 21
41 Definability Second level So on which level do things go wrong? Question: Does the assertion all Σ 1 2 sets are regular hold? Answer: It is independent of ZFC! Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 11 / 21
42 Independence results 3. Independence results Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 12 / 21
43 Independence results Constuctible universe and extensions L = Gödel s constructible universe. Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 13 / 21
44 Independence results Constuctible universe and extensions L = Gödel s constructible universe. There is a Σ 1 2-definable well-ordering of the continuum. Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 13 / 21
45 Independence results Constuctible universe and extensions L = Gödel s constructible universe. There is a Σ 1 2-definable well-ordering of the continuum. Therefore irregularity exists on the Σ 1 2 (even 1 2) level. Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 13 / 21
46 Independence results Constuctible universe and extensions L = Gödel s constructible universe. There is a Σ 1 2-definable well-ordering of the continuum. Therefore irregularity exists on the Σ 1 2 (even 1 2) level. Forcing over L. By forcing we can add new reals, destroy Σ 1 2 well-ordering. Does irregularity disappear? Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 13 / 21
47 Independence results Constuctible universe and extensions L = Gödel s constructible universe. There is a Σ 1 2-definable well-ordering of the continuum. Therefore irregularity exists on the Σ 1 2 (even 1 2) level. Forcing over L. By forcing we can add new reals, destroy Σ 1 2 well-ordering. Does irregularity disappear? If we add many reals, yes. If we add not so many reals, perhaps not. Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 13 / 21
48 Independence results Constuctible universe and extensions L = Gödel s constructible universe. There is a Σ 1 2-definable well-ordering of the continuum. Therefore irregularity exists on the Σ 1 2 (even 1 2) level. Forcing over L. By forcing we can add new reals, destroy Σ 1 2 well-ordering. Does irregularity disappear? If we add many reals, yes. If we add not so many reals, perhaps not. In fact, we can say exactly which reals must be added to obtain regularity on Σ 1 2/ 1 2 level. Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 13 / 21
49 Independence results Solovay-Judah-Shelah characterizations Theorem (Judah-Shelah 1989) The following are equivalent: 1 All 1 2 sets are Lebesgue-measurable, 2 For all a R there is a random-generic real over L[a]. Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 14 / 21
50 Independence results Solovay-Judah-Shelah characterizations Theorem (Judah-Shelah 1989) The following are equivalent: 1 All 1 2 sets are Lebesgue-measurable, 2 For all a R there is a random-generic real over L[a]. Theorem (Solovay 1969) The following are equivalent: 1 All Σ 1 2 sets are Lebesgue-measurable, 2 For all a R, almost all reals are random-generic over L[a]. Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 14 / 21
51 Independence results Solovay-Judah-Shelah characterizations Theorem (Judah-Shelah 1989) The following are equivalent: 1 All 1 2 sets have the Baire property, 2 For all a R there is a Cohen-generic real over L[a]. Theorem (Solovay 1969) The following are equivalent: 1 All Σ 1 2 sets have the Baire property, 2 For all a R, almost all reals are Cohen-generic over L[a]. Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 15 / 21
52 Independence results Iterated forcing extensions Statements all Σ 1 2 ( 1 2) sets are regular correspond to transcendence over L. Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 16 / 21
53 Independence results Iterated forcing extensions Statements all Σ 1 2 ( 1 2) sets are regular correspond to transcendence over L. Since transcendence over L can (to some extend) be controlled by forcing, so can regularity. Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 16 / 21
54 Independence results Iterated forcing extensions Statements all Σ 1 2 ( 1 2) sets are regular correspond to transcendence over L. Since transcendence over L can (to some extend) be controlled by forcing, so can regularity. Example 1. Random forcing adds random-generic reals but not Cohen-generic reals. Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 16 / 21
55 Independence results Iterated forcing extensions Statements all Σ 1 2 ( 1 2) sets are regular correspond to transcendence over L. Since transcendence over L can (to some extend) be controlled by forcing, so can regularity. Example 1. Random forcing adds random-generic reals but not Cohen-generic reals. Therefore, if we iterate random forcing for ℵ 1 steps, we get a model where all 1 2 sets are Lebesgue measurable, but not all 1 2 sets have the Baire property. Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 16 / 21
56 Independence results Iterated forcing extensions Statements all Σ 1 2 ( 1 2) sets are regular correspond to transcendence over L. Since transcendence over L can (to some extend) be controlled by forcing, so can regularity. Example 1. Random forcing adds random-generic reals but not Cohen-generic reals. Therefore, if we iterate random forcing for ℵ 1 steps, we get a model where all 1 2 sets are Lebesgue measurable, but not all 1 2 sets have the Baire property. Example 2. Cohen forcing adds Cohen-generic reals but not random-generic reals. Therefore, if we iterate Cohen forcing (for ℵ 1 steps), we get a model where all 1 2 sets have the Baire property but not all 1 2 sets are Lebesgue measurable. Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 16 / 21
57 Independence results Strength of measurability On the other hand, some properties are stronger than others: Theorem (Bartoszyński-Raisonnier-Stern 1984/1985) If all Σ 1 2 sets are Lebesgue measurable then all Σ 1 2 sets have the Baire property. Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 17 / 21
58 Independence results Strength of measurability On the other hand, some properties are stronger than others: Theorem (Bartoszyński-Raisonnier-Stern 1984/1985) If all Σ 1 2 sets are Lebesgue measurable then all Σ 1 2 sets have the Baire property. Measurability statements have various strength, corresponding to strength of transcendence statements. Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 17 / 21
59 Independence results Strength of measurability On the other hand, some properties are stronger than others: Theorem (Bartoszyński-Raisonnier-Stern 1984/1985) If all Σ 1 2 sets are Lebesgue measurable then all Σ 1 2 sets have the Baire property. Measurability statements have various strength, corresponding to strength of transcendence statements. Σ 1 2(Lebesgue) Σ 1 2(Baire) 1 2(Lebesgue) 1 2(Baire) Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 17 / 21
60 Independence results Brendle & Löwe, Eventually different functions and inaccessible cardinals. Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 18 / 21
61 Independence results Σ 1 2 (Ramsey) Σ 1 2 (Laver) a dom./l[a] Σ 1 2 (Miller) a unb./l[a] Σ 1 2 ( n m) (a) (b) (c ) Σ 1 2 ( ω m) 1 2 ( n m) (b ) (c) (d ) 1 2 ( ω m) (b ) (d) a ev. diff./l[a] a bounded ev. diff./l[a] (e ) Brendle & Khomskii, Polarized partitions on the second level of the projective hierarchy. (e) a {x x not I E0 - quasigeneric /L[a]} N E0 a I E0 -quasigeneric/l[a] 1 2 (doughnut) a splitting/l[a] Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 19 / 21
62 Independence results Questions Typical questions in this field: Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 20 / 21
63 Independence results Questions Typical questions in this field: 1 Given a regularity property, characterize it by transcendence property. Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 20 / 21
64 Independence results Questions Typical questions in this field: 1 Given a regularity property, characterize it by transcendence property. 2 Given a transcendence property, characterize it by regularity. Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 20 / 21
65 Independence results Questions Typical questions in this field: 1 Given a regularity property, characterize it by transcendence property. 2 Given a transcendence property, characterize it by regularity. 3 Find general Solovay-Judah-Shelah-style theorems (some work done by Daisuke Ikegami; still many open questions). Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 20 / 21
66 Independence results Questions Typical questions in this field: 1 Given a regularity property, characterize it by transcendence property. 2 Given a transcendence property, characterize it by regularity. 3 Find general Solovay-Judah-Shelah-style theorems (some work done by Daisuke Ikegami; still many open questions). 4 Prove implications from Σ 1 2/ 1 2(Reg 1 ) to Σ 1 2/ 1 2(Reg 2 ), or produce a model which separates Reg 1 from Reg 2. Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 20 / 21
67 Independence results Questions Typical questions in this field: 1 Given a regularity property, characterize it by transcendence property. 2 Given a transcendence property, characterize it by regularity. 3 Find general Solovay-Judah-Shelah-style theorems (some work done by Daisuke Ikegami; still many open questions). 4 Prove implications from Σ 1 2/ 1 2(Reg 1 ) to Σ 1 2/ 1 2(Reg 2 ), or produce a model which separates Reg 1 from Reg 2. 5 For some properties, whether it holds on the Σ 1 1 or even Borel level is still open (e.g., does there exist a Borel maximal family of eventually different functions?) Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 20 / 21
68 Independence results Thank you! Yurii Khomskii Yurii Khomskii (University of Amsterdam) Regularity and definability PhDs in Logic III 21 / 21
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