The Theory of Higher Randomness

The Theory of Higher Randomness Liang Yu Department of mathematics National University of Singapore Joint with CT Chong and A. Nies

12th July 2006

History

Spector’s measure-theoretic construction of two incomparable hyperdegrees. Sacks’s result that every nontrivial cone of hyperdegrees is null. Measure-theoretic uniformity in recursion theory and set theory. ¨ result on ∆11 randomness. Martin-Lof’s

Motivation

The genuine randomness; Characterizing these reals; More applications of measure theoretic argument to recursion theory and set theory.

The language L(ω1CK , §)

1

Number variables:j, k , m, n, ...;

2

Numerals:0,1,2,...;

3

Ranked real variables: x α , y α where α < ω1CK ; ˙ y, y, ˙ ...; Unranked real variables: x, x,

4 5

Others: +, · (times), 0 (successor) and ∈.

The number theoretic terms are: numerals, number variables, n + m, n · m and n0 . The atomic formulas are n = m and n ∈ x. Formulas are built by the usual way. A formula ϕ is ranked if all of its real variables are ranked.

Coding the language

A uniform way coding the language: ¨ Fix a Π11 path O1 through O. The Godel number of x α is (2, n) where n ∈ O1 and |n| = α. A formula ϕ is Σ11 if it is of the form ∃x1 ∃x2 , ...∃xn ψ where ψ is a ranked formula. ¨ Note that {nϕ |ϕ is Σ11 } is a Π11 set where nϕ is the Godel number of ϕ.

The ramified analytic hierarchy

A(α, x) consists of those sets first order definable over A(β, x) for all β < α.S A(ω1CK , x) = α p} is Π11 . Moreover, for each Π11 set A ⊆ 2ω , there is a formula ϕ ∈ Σ11 ˙ implies x ∈ A. Moreover, if ω1x = ω1CK , so that A(ω1CK , x) |= ϕ(x) CK ˙ then x ∈ A implies A(ω1 , x) |= ϕ(x). Theorem (Sacks and Tanaka) If A is Π11 and has positive measure, then A contains a hyperarithmetical real.

Theorem (Sacks) There is a recursive function f : ω × Q → ω so that for all n ¨ which is a Godel number of ranked formula 1 ¨ f (n, p) is a Godel number of a ranked formula; 2

The set {x|A(ω1CK , x) |= ϕf (n,p) } ⊇ {x|A(ω1CK , x) |= ϕn } is open;

3

µ({x|A(ω1CK , x) |= ϕf (n,p) } − {x|A(ω1CK , x) |= ϕn }) < p.

Theorem (Sacks) The set {x|ω1x > ω1CK } = {x|x ≥h O} is null.

Higher randomness

¨ test A sequence open sets {Un }n∈ω is said to be a Martin-Lof (ML-test) if µ(Un ) ≤ 2−n for all n. Definition Given a class of sets of reals Γ, 1 2

A real x is Γ random if no Γ null set contains x. T A real x is Γ-ML-random if x 6∈ n∈ω Vn for any ML-test {Vn }n so that {(n, x)|x ∈ Vn } ∈ Γ.

∆11 -randomness

Theorem (Sacks) ∆11 -ML-randomness=∆11 -randomness=Σ11 -randomness. ¨ Theorem (Marti-Lof) The set {x|x is a ∆11 − random real} is Π11 .

Π11 -ML-randomness

Theorem (Hjorth and Nies) 1 2

3

There exists a universal Π11 -ML test. Π11 -random reals range over all of the hyperdegrees above O. There exists a proper Π11 real which is KΠ1 -trivial. 1

4

If ω1x = ω1CK , then x is KΠ1 -trivial if and only if x is 1 hyperarithmetic.

Note that Sacks (implicitly) proved that there exists no largest Σ11 set.

Π11 -randomness

Theorem (Kechris; Hjorth and Nies) There exists the largest Π11 null set. Proof. ˙ ˙ ∧ µ({x|A(ω1CK , x)  ¬ϕn (x)}) ≥ 1)} Qn = {x|A(ω1CK , x)  ϕn (x) ¨ where n ranges over the Godel numbers of ranked formula, and [ Q= Qn ∪ {x|ω1x > ω1CK }. n∈ω

Corollary If ω1x = ω1CK , then x is ∆11 -random if and only if x is Π11 -ML random if and only if x is Π11 -random. Observations: If x is Π11 -random, then ω1x = ω1CK ; There exists a Π11 -random real hyper-reducible to O; {x|ω1x > ω1CK } is a ∆11 (O)-set.

Summary

Theorem ∆11 (O)-randomness ⊂ Π11 -randomness ⊂ Π11 -ML randomness ⊂ ∆11 -randomness = ∆11 -ML randomness.

Hyperimmune-freeness and traceability

Definition A real x is HYP-hyperimmunefree (HYP-hif) if for all function f : ω → ω with f ≤h x, there is a hyperarithmetic function g so that g(n) > f (n) for all n (i.e. g > f ). Definition Fix a recursive enumeration {Dn }n∈ω of finite sets of numbers. A real x is HYP-traceable if there is a ∆11 -function h : ω → ω so that for all function f : ω → ω with f ≤h x, there is a hyperarithmetic function g : ω → ω so that for all n: 1

|Dg(n) | ≤ h(n);

2

f (n) ∈ Dg(n) .

Basic facts

Proposition {x|x is HYP-hif } ⊂ {x|ω1x = ω1CK }. Theorem There are 2ℵ0 -many HYP-traceable reals.

Higher randomness vs. randomness

Theorem Each 2-random real is not hif. Each Π11 -random real is HYP-hif. Proposition Each non-empty Π01 set of reals contains a hif real. There is a non-empty Σ11 set A ⊆ 2ω which does not contain HYP-hif real.

Lowness

Definition Given a notation G with relativized version Gx . A real x is said to be low for G if G is the same as Gx .

Lowness for randomness

Theorem A real is low for ∆11 -randomness if and only if it is hyperarithmetical traceable. Theorem (Hjorth and Nies) x is low for Π11 -ML randomness if and only if x is hyperarithmetic. Proposition (Hjorth and Nies) If x is low for Π11 -randomness, then ω1CK = ω1x .

An open question

Question Is there a non-hyperarithmetical real x low for Π11 -randomness?

Beyond ZFC

1

Projective determinacy;

2

Fine structure.

Thank you