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159
Formalizing forcing arguments in subsystems of secondorder arithmetic
 Annals of Pure and Applied Logic
, 1996
"... We show that certain modeltheoretic forcing arguments involving subsystems of secondorder arithmetic can be formalized in the base theory, thereby converting them to effective prooftheoretic arguments. We use this method to sharpen conservation theorems of Harrington and BrownSimpson, giving an ..."
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We show that certain modeltheoretic forcing arguments involving subsystems of secondorder arithmetic can be formalized in the base theory, thereby converting them to effective prooftheoretic arguments. We use this method to sharpen conservation theorems of Harrington and BrownSimpson, giving an effective proof that W KL+0 is conservative over RCA0 with no significant increase in the lengths of proofs. 1
Automorphisms of the lattice of Π 0 1 classes: perfect thin classes and anc degrees
 Trans. Amer. Math. Soc
"... Abstract. Π0 1 classes are important to the logical analysis of many parts of mathematics. The Π0 1 classes form a lattice. As with the lattice of computably enumerable sets, it is natural to explore the relationship between this lattice and the Turing degrees. We focus on an analog of maximality, o ..."
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Abstract. Π0 1 classes are important to the logical analysis of many parts of mathematics. The Π0 1 classes form a lattice. As with the lattice of computably enumerable sets, it is natural to explore the relationship between this lattice and the Turing degrees. We focus on an analog of maximality, or more precisely, hyperhypersimplicity, namely the notion of a thin class. We prove a number of results relating automorphisms, invariance, and thin classes. Our main results are an analog of Martin’s work on hyperhypersimple sets and high degrees, using thin classes and anc degrees, and an analog of Soare’s work demonstrating that maximal sets form an orbit. In particular, we show that the collection of perfect thin classes (a notion which is definable in the lattice of Π0 1 classes) forms an orbit in the lattice of Π01 classes; and a degree is anc iff it contains a perfect thin class. Hence the class of anc degrees is an invariant class for the lattice of Π0 1 classes. We remark that the automorphism result is proven via a ∆0 3 automorphism, and demonstrate that this complexity is necessary. 1.
Degrees of unsolvability of continuous functions
 Journal of Symbolic Logic
"... Abstract. We show that the Turing degrees are not sufficient to measure the complexity of continuous functions on [0, 1]. Computability of continuous real functions is a standard notion from computable analysis. However, no satisfactory theory of degrees of continuous functions exists. We introduce ..."
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Abstract. We show that the Turing degrees are not sufficient to measure the complexity of continuous functions on [0, 1]. Computability of continuous real functions is a standard notion from computable analysis. However, no satisfactory theory of degrees of continuous functions exists. We introduce the continuous degrees and prove that they are a proper extension of the Turing degrees and a proper substructure of the enumeration degrees. Call continuous degrees which are not Turing degrees nontotal. Several fundamental results are proved: a continuous function with nontotal degree has no least degree representation, settling a question asked by PourEl and Lempp; every noncomputable f ∈ C[0, 1] computes a noncomputable subset of N; there is a nontotal degree between Turing degrees a <T b iff b is a PA degree relative to a; S ⊆ 2N is a Scott set iff it is the collection of fcomputable subsets of N for some f ∈ C[0, 1] of nontotal degree; and there are computably incomparable f, g ∈ C[0, 1] which compute exactly the same subsets of N. Proofs draw from classical analysis and constructive analysis as well as from computability theory. §1. Introduction. The computable real numbers were introduced in Alan Turing’s famous 1936 paper, “On computable numbers, with an application to the Entscheidungsproblem ” [40]. Originally, they were defined to be the reals
The Baire category theorem in weak subsystems of secondorder arithmetic
 THE JOURNAL OF SYMBOLIC LOGIC
, 1993
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The metamathematics of ergodic theory
 THE ANNALS OF PURE AND APPLIED LOGIC
, 2009
"... The metamathematical tradition, tracing back to Hilbert, employs syntactic modeling to study the methods of contemporary mathematics. A central goal has been, in particular, to explore the extent to which infinitary methods can be understood in computational or otherwise explicit terms. Ergodic theo ..."
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The metamathematical tradition, tracing back to Hilbert, employs syntactic modeling to study the methods of contemporary mathematics. A central goal has been, in particular, to explore the extent to which infinitary methods can be understood in computational or otherwise explicit terms. Ergodic theory provides rich opportunities for such analysis. Although the field has its origins in seventeenth century dynamics and nineteenth century statistical mechanics, it employs infinitary, nonconstructive, and structural methods that are characteristically modern. At the same time, computational concerns and recent applications to combinatorics and number theory force us to reconsider the constructive character of the theory and its methods. This paper surveys some recent contributions to the metamathematical study of ergodic theory, focusing on the mean and pointwise ergodic theorems and the Furstenberg structure theorem for measure preserving systems. In particular, I characterize the extent to which these theorems are nonconstructive, and explain how prooftheoretic methods can be used to locate their “constructive content.”
The Medvedev lattice of computably closed sets
"... Simpson introduced the lattice P of 0 1 classes under Medvedev reducibility. Questions regarding completeness in P are related to questions about measure and randomness. We present a solution to a question of Simpson about Medvedev degrees of 0 1 classes of positive measure that was independently ..."
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Simpson introduced the lattice P of 0 1 classes under Medvedev reducibility. Questions regarding completeness in P are related to questions about measure and randomness. We present a solution to a question of Simpson about Medvedev degrees of 0 1 classes of positive measure that was independently solved by Simpson and Slaman. We then proceed to discuss connections to constructive logic. In particular we show that the dual of P does not allow an implication operator (i.e. that P is not a Heyting algebra). We also discuss properties of the class of PAcomplete sets that are relevant in this context.
Effective Search Problems
 Mathematical Logic Quarterly
, 1994
"... The task of computing a function F with the help of an oracle X can be viewed as a search problem where the cost measure is the number of queries to X . We ask for the minimal number that can be achieved by a suitable choice of X and call this quantity the query complexity of F . This concept is s ..."
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The task of computing a function F with the help of an oracle X can be viewed as a search problem where the cost measure is the number of queries to X . We ask for the minimal number that can be achieved by a suitable choice of X and call this quantity the query complexity of F . This concept is suggested by earlier work of Beigel, Gasarch, Gill, and Owings on "Bounded query classes". We introduce a fault tolerant version and relate it with Ulam's game. For many natural classes of functions F we obtain tight upper and lower bounds on the query complexity of F . Previous results like the Nonspeedup Theorem and the Cardinality Theorem appear in a wider perspective. 1991 Mathematics Subject Classification: Primary 03D20; Secondary 68Q15, 68R05 Keywords: Search problems, bounded queries, query complexity, recursive functions 1 Introduction The task of computing a function F with the help of an oracle X ` ! (! is the set of all natural numbers) can be viewed as a search problem where t...
ComputabilityTheoretic and ProofTheoretic Aspects of Partial and Linear Orderings
 Israel Journal of mathematics
"... Szpilrajn's Theorem states that any partial order P = hS; <P i has a linear extension L = hS; <L i. This is a central result in the theory of partial orderings, allowing one to de ne, for instance, the dimension of a partial ordering. It is now natural to ask questions like \Does a we ..."
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Szpilrajn's Theorem states that any partial order P = hS; <P i has a linear extension L = hS; <L i. This is a central result in the theory of partial orderings, allowing one to de ne, for instance, the dimension of a partial ordering. It is now natural to ask questions like \Does a wellpartial ordering always have a wellordered linear extension?" Variations of Szpilrajn's Theorem state, for various (but not for all) linear order types , that if P does not contain a subchain of order type , then we can choose L so that L also does not contain a subchain of order type . In particular, a wellpartial ordering always has a wellordered extension.
DEMUTH RANDOMNESS AND COMPUTATIONAL COMPLEXITY
"... Demuth tests generalize MartinLöf tests (Gm)m∈N in that one can exchange the mth component for a computably bounded number of times. A set Z ⊆ N fails a Demuth test if Z is in infinitely many final versions of the Gm. If we only allow Demuth tests such that Gm ⊇ Gm+1 for each m, we have weak Demu ..."
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Demuth tests generalize MartinLöf tests (Gm)m∈N in that one can exchange the mth component for a computably bounded number of times. A set Z ⊆ N fails a Demuth test if Z is in infinitely many final versions of the Gm. If we only allow Demuth tests such that Gm ⊇ Gm+1 for each m, we have weak Demuth randomness. We show that a weakly Demuth random set can be high, yet not superhigh. Next, any c.e. set Turing below a Demuth random set is strongly jumptraceable. We also prove a basis theorem for nonempty Π 0 1 classes P. It extends the JockuschSoare basis theorem that some member of P is computably dominated. We use the result to show that some weakly 2random set does not compute a 2fixed point free function.
THE STRENGTH OF THE RAINBOW RAMSEY THEOREM
, 2009
"... The Rainbow Ramsey Theorem is essentially an “antiRamsey” theorem which states that certain types of colorings must be injective on a large subset (rather than constant on a large subset). Surprisingly, this version follows easily from Ramsey’s Theorem, even in the weak system RCA0 of reverse mathe ..."
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The Rainbow Ramsey Theorem is essentially an “antiRamsey” theorem which states that certain types of colorings must be injective on a large subset (rather than constant on a large subset). Surprisingly, this version follows easily from Ramsey’s Theorem, even in the weak system RCA0 of reverse mathematics. We answer the question of the converse implication for pairs, showing that the Rainbow Ramsey Theorem for pairs is in fact strictly weaker than Ramsey’s Theorem for pairs over RCA0. The separation involves techniques from the theory of randomness by showing that every 2random bounds an ωmodel of the Rainbow Ramsey Theorem for pairs. These results also provide as a corollary a new proof of Martin’s theorem that the hyperimmune degrees have measure one.