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<ωCK A(α, x). 1
Basic facts on ramified analytic hierarchy
Theorem (Sacks) TFAE: 1
ω1CK = ω1x ;
2
A(ω1CK , x) = {y|y ≤h x};
3
A(ω1CK , x) satisfies ∆11 -comprehension.
Effective measure theory
Theorem (Sacks) The set {(nϕ , p)|p ∈ Q ∧ ϕ ∈ Σ11 ∧ µ({x|A(ω1CK , x) |= ϕ}) > 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