Chitat Chong

The Metamathematics Of Stable Ramsey’s Theorem For Pairs

We show that, over the base theory RCA0, Stable Ramseys The- orem for Pairs implies neither Ramseys Theorem for Pairs nor � 0-induction.

On the role of the collection principle for Sigma^0_2-formulas in second-order reverse mathematics

We show that the principle PART from Hirschfeldt and Shore is equivalent to the ∑0 2 -Bounding principle B∑ 0 2 over RCA 0 , answering one of their open questions. Furthermore, we also fill a gap in a proof of Cholak, Jockusch and Slaman by showing that D 2 2 implies B∑ 0 2 and is thus indeed equivalent to Stable Ramseys Theorem for Pairs (SRT 2 2 ). This also allows us to conclude that the combinatorial principles IPT 2 2 , SPT 2 2 and SIPT 2 2 defined by Dzhafarov and Hirst all imply B∑ 0 2 and thus that SPT 2 2 and SIPT 2 2 are both equivalent to SRT 2 2 as well. Our proof uses the notion of a bi-tame cut, the existence of which we show to be equivalent, over RCA 0 , to the failure of B∑ 0 2 .

The degree of a Σn cut

Let P − denote the Peano axioms minus the induction scheme. IΣ n and BΣ n denote respectively the Σ n induction scheme and the Σ n collection scheme. A theorem of Paris and Kirby states that P − +IΣ n implies P − +BΣ n implies P − +IΣ n−1 , but not conversely. This justifies investigations into models of fragments of Peano arithmetic from the recursion-theoretic point of view. We study more closely the degree of cuts in models of P − +BΣ n and how they relate to the structure of degrees. We are in particular interested in problems on the existence of degrees whose original proofs depend on the assumption of at least IΣ 2

∑2 Induction and infinite injury priority arguments, part II Tame ∑2 coding and the jump operator

The jump operator ’ is a fundamental operation in recursion theory. In a previous paper [3], we considered the jump of an r.e. set in subsystems of Peano arithmetic. In any model of C2 induction, it is known that the Sacks Jump Inversion Theorem holds, so that any degree r.e. in and above 0’ is the jump of an r.e. degree. An r.e. set is high if its jump is of degree 0”. The main theorem of [3] in this connection states that the existence of an incomplete high r.e. set is equivalent to C2 induction over the base theory P(the basic arithmetic operations) plus Cz collection (B&). This result characterizes the proof-theoretic complexity of the existence of such r.e. sets, and elucidates the (necessary) use of infinite injury priority arguments in such constructions. The question of the jump of an r.e. set remained: In a model without C2 induction, what are the degrees of A’ for an r.e. set A ? It turns out that this question is closely related to the existence of codes for bounded sets. In a model of P+ BC2 without C2 induction, call a cut C2 if it is &-definable. A subset X of a cut Z is coded on Z if there is a finite set (in the sense of the model) whose intersection with Z is equal to X. Mytilinaios and Slaman [9] first pointed out how for some structures, the existence of codes for reals could affect the jump of r.e. sets. They constructed a model in which every real was coded on o, with o acting as a C2 cut. In this model 0’ is the jump of every incomplete r.e. set. On the other hand, we introduced in [3] a model, with CLI as a C2 cut and having a &-definable real not coded on 0, such that a > 0’ for some r.e. degree a. These results hinted at a link between the existence of codes and the degrees of jumps of r.e. sets.

Degrees bounding minimal degrees

A set is called n -generic if it is Cohen generic for n -quantifier arithmetic. A (Turing) degree is n -generic if it contains an n -generic set. Our interest in this paper is the relationship between n -generic (indeed 1-generic) degrees and minimal degrees, i.e. degrees which are non-recursive and which bound no degrees intermediate between them and the recursive degree. It is known that n -generic degrees and minimal degrees have a complex relationship since Cohen forcing and Sacks forcing are mutually incompatible. The goal of this paper is to show.

Randomness in the higher setting

We study the strengths of various notions of higher randomness: (i) strong Π1-ML-randomness is separated from Π 1 1-ML-randomness; (ii) the hyperdegrees of Π1-random reals are closed downwards (except for the trivial degree); (iii) the reals z in NCRΠ11 are precisely those satisfying z ∈ Lωz 1 , and (iv) lowness for ∆1-randomness is strictly weaker than that for Π 1 1-randomness.

∑2 Induction and infinite injury priority arguments, part III: Prompt sets, minimal pairs and Shoenfield’s Conjecture

We prove that in everyB∑2 model (one satisfies ∑2 collection axioms but not ∑2 induction), every recursively enumerable (r.e.) set is either prompt or recursive. Consequently, over the base theory ∑2 collection, the existence of r.e. minimal pairs is equivalent to ∑2 induction. We also refute Shoenfield’s Conjecture inB∑2 models.

Major subsets ofα-recursively enumerable sets

Let s2cf(α), s2p(α) and ts2p(α) denote the Σ2-confinality, Σ2-projectum and the tame Σ2-projectum of an admissible ordinalα. We show that if s2cf(α)<s2p(α), then noα-recursively enumerable set (α-r.e.) with complement of order type less than ts2p(α) can have a major subset. As a corollary, if s2cf(α)<s2p(α), then no hyperhypersimpleα-r.e. set can have a major subset.

A

We introduce a Π 1 1 -uniformization principle and establish its equivalence with the set-theoretic hypothesis (ω 1 ) L = ω 1 . This principle is then applied to derive the equivalence, to suitable set-theoretic hypotheses, of the existence of Π 1 1 -maximal chains and thin maximal antichains in the Turing degrees. We also use the Π 1 1 -uniformization principle to study Martins conjectures on cones of Turing degrees, and show that under V = L the conjectures fail for uniformly degree invariant Π 1 1 functions.

\Pi ^1_1

Emil Post, in his attempts to prove the existence of an incomplete recursively enumerable (t.e.) degree, introduced [12] the notion of a hyperhypersimple @h-simple) set and deEned it to be a set W for which there is no recursive sequence of pairwise disjoint r.e. sets {He}cC, such that Over the standard model of arithmetic N, this definition is equivalent (Lachlan [S]) to the assertion that the lattice of r.e. supersets of A forms a Boolean algebra. There is an obvious advantage of adopting this latter form as the definition for hyperhypersimplic@ since it is lattice theoretic and so, as the works of various authors have shown, provide an important avenue of study for the decision problem of the lafiice of r.e. sets. In this paper we will adopt this lattice theoretic definition of hyperhypersimplicitit): and studY the problem of existence of hh-simple sets for various models of fragments of Peano arithmetic. Our interest is primarilY proof-theoretic, i.e. to fmd necessargr and/or sufficient conditions in the hierarchy of induction schema for the existence of these sets. Thus the work may be considered to fall within the realm of a very general question in what one might call reverse recursion theory: at axioms of Peano arithmetic are necessary or sufficient to prove theorems in recursion theory? This question (perhaps first raised by Stephen Simpson) is a natural offshoot of a related, more general question: Which set existence axioms of second order arithmetic are required, or sufficient, to prove theorems in ordinary mathematics (Simpson M) ? A second motivation for our interest in this sujbect is methodological. Recent works in this area (for example linaios & Slaman [ 111, Groszek & Slaman [S], Slaman & odin [ 141, Thong [ 11) point to the strong connection of this subject with higher recursion theory. In particular, many techniques orig: .lally invented to tackle problems in ordinal recursion have been successfully adapted to solve problems here. The fact that ordinal recursion out to be relevant to classical recursion theory in an important way is perhaps an unexpected development, althoug isel in his paper [7] had me reasons for generalizing recursion theory, and he and need to understand classical recursion theory better Yo ecent development in