D-R.E Degrees and Fragments of Arithmetic

Yang Yue

Department of MathematicsNational University of Singapore

May 12, 2013

Outline

Structure of d-r.e. degrees

D-R.E Degrees in fragments of Arithmetic

∆02-Sets

Limit Lemma (Shoenfield)Let A ⊆ N. TFAE:(1) A ≤T ∅′.

(2) A is ∆02.

(3) There is a recursive f (x , s) s.t. for all x lims

f (x , s) = A(x).

n-r.e. sets

DefinitionA set A ⊆ N is called n-r.e. iff there is a recursive f (x , s) s.t. forall x(a) f (x ,0) = 0;(b) lim

sf (x , s) = A(x);

(c) |{s : f (x , s) 6= f (x , s + 1)}| ≤ n.

E.g., 1-r.e. is just r.e.; and 2-r.e. is d-r.e (difference of two r.e.sets).

Proper d-r.e. sets exist and first studied by Putnam and Gold in1965.

n-R.E degrees

I Theorem (Cooper 1971) There is a proper d-r.e. degree.

I The proof uses finite injury argument (generalize to getproper n-r.e. degrees).

I They form a proper hierarchy:

R ( D2 ( D3 ( · · · ( D(≤ 0′).

I Q: Are they elementarily equivalent?

Elementary Equivalence Problems

I A ≡k B iff for any Σk -sentence τ , |=A τ ⇔|=B τ .

I A ≡ B iff for all k , A ≡k B.

I Assume |A| ⊆ |B|. A 4k B iff for any Σk -sentence τ withparameters from A, |=A τ ⇔|=B τ .

I A 4 B iff for all k , A 4k B.

No Differences at Σ1-level

Theorem (first observed by Sacks)

R ≡1 D2 ≡1 D3 ≡1 · · · ≡1 D(≤ 0′).

Essentially from Friedberg-Muchnik Theorem, we can embedall countable partial orders.

τ is true in any one of the structures iff τ is consistent with thetheory of partial orders.

D(≤ 0′) 6≡2 the rest

Theorem (Sacks, 1961)There is a minimal degree ≤T 0′.

Theorem (Sacks, 1964)R is dense.

Theorem (Lachlan)Dn is downward dense, in fact, for all nonzero d ∈ Dn there is anonzero r.e. a < d.

R 6≡2 the rest

We have seen R is dense.

Theorem (Cooper, Harrington, Lachlan, Lempp andSoare, 1991)There is a maximal d in D2; in fact no ω-r.e. a with d < a < 0′.

Some History

I The first difference between R and Dn (n ≥ 2) wasdiscovered by Arslanov:

Theorem (Arslanov, 1985)Every d 6= 0 in Dn is cuppable in Dn.

I Coupled with

Theorem (Cooper and Yates, 1973)There is noncuppable r.e. degree.

We see a Σ3-difference.

Some History: Downey’s Conjecture

Theorem (Downey, 1989)There is a diamond in D2.

I Coupled with

Theorem (Lachlan, 1966)There is no diamond in R.

We see a Σ2-difference.

Downey Conjecture (1989)For all n,m ≥ 2, Dn ≡ Dm.

D2 6≡2 D3

Theorem (Arslanov, Kalimullin and Lempp, 2010)D2 6≡2 D3.

Conjecture (AKL) Dn 6≡ Dm.

Proof by Bubbles

Proof.“four bubbles” can be embedded into D3 but not D2.

(Perhaps the AKL conjecture can be proved by more bubbles?)

Remaining Questions

I Seen: Differences start showing up at Σ2-level.

I Q: Is one structure a Σ1-elementary substructure of theother? I.e., can we pick out the difference at Σ1-level usingparameters?

Slaman TriplesTheorem (Slaman, 1983)R 641 D(≤ 0′).

The proof used Slaman triples:

Slaman Theorem

Theorem (Slaman)There are r.e. degrees a,b and c, and an x ≤ 0′ such that(a) a 6= 0;(b) c � b;(c) for all r.e. degree w ≤ a either w = 0 or w ∨ b ≥ c;(d) x ≤ a, x 6= 0 and x ∨ b � c.

CorollaryDn 641 D(≤ 0′).

R 641 D2

Theorem (Yang and Yu, 2006)R 641 D2.

Notice that the x is Slaman’s theorem cannot be in Dn, byLachlan’s lower density result.

The idea is to introduce another parameter.

A Picture

A Technical Theorem

Theorem (Yang and Yu, 2006)There are r.e. degrees a,b,c and e, and a d ∈ D2 such that(a) a 6= 0;(b) c � b;(c) for all r.e. degree w ≤ a either w ≤ e or w ∨ b ≥ c;(d) d ≤ a, d � e and d ∨ b � c.

CorollaryR 641 Dn.

Recent Progress

Theorem (Cai, Shore and Slaman, 2012)For all n < m, Dn 641 Dm.

They also showed that the first order theory of Dn isundecidable.

Open question: Is Th(Dn) isomorphic to first order arithmetic?

[Known: Th(R) and Th(D(≤ 0′)) are.]

Motivations of Reverse Recursion Theory

I In Recursion Theory, the constructions are verified byinduction; in particular, in priority arguments.

I (Similar to reverse math) How much first induction does itrequire to prove a theorem?

I Another motivation: Studying computability in moregeneral domains, like in α-recursion theory.

Fragments of Peano Arithmetic

I Let IΣn denote the induction schema for Σ0n-formulas; and

BΣn denote the Bounding Principle for Σ0n formulas.

I (Kirby and Paris, 1977) · · · ⇒ IΣn+1 ⇒ BΣn+1 ⇒ IΣn ⇒ . . .

I (Slaman 2004) I∆n ⇔ BΣn.

Recursion Theory onM

I A set A ⊂ M is r.e iff A is Σ01-definable inM with

parameters.

I A set A is recursive iff A and M \ A are r.e.

I So we can study recursion theory on weak fragments ofPeano arithmetic.

Sample Results in Reverse Recursion

I Over PA− + BΣ1: IΣ1 ⇔ Existence of low r.e. sets⇔Sacks Splitting Theorem

I Over PA− + BΣ2: IΣ2 ⇔ Existence of high r.e. sets⇔Minimal Pair Theorem

I As in Reverse Mathematics, new proofs are required whenworking in fragments.

Reversing Cooper

I IΣ1 proves Cooper’s theorem.

I Theorem (Li Wei) BΣ1 proves existence of properd-r.e. degrees (but not below 0′).

I In fact, she showed that in every BΣ1-model, alld-r.e. degrees below 0′ are r.e.

I Furthermore in saturated BΣ1-models, all degrees below0′ are r.e.

Isolation Pairs

I One can study the “relative” structure R within D2.

I An r.e. degree a isolates a d-r.e. degree d if a < d but forall r.e. degree w < d , w < a.

I Theorem (Cooper and Yi; Kaddah) Such isolation pairexists.

D-C.E and DC.D

I D-r.e. is also known as d-c.e.

I Theorem (D.C Ding and Qian, 1996) Isolation pairs aredense in r.e. degrees.

I Theorem (Ishmukhametov and Wu) There are low-highisolation pairs.

Work in Progress

I The construction of isolation pair is sort of between finiteand infinite injury.

I Claim (Liu Yiqun) IΣ1 proves the existence of isolation pair.