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On the strength of Ramsey’s Theorem for pairs
 Journal of Symbolic Logic
, 2001
"... Abstract. We study the proof–theoretic strength and effective content denote Ramof the infinite form of Ramsey’s theorem for pairs. Let RT n k sey’s theorem for k–colorings of n–element sets, and let RT n < ∞ denote (∀k)RTn k. Our main result on computability is: For any n ≥ 2 and any computable (r ..."
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Cited by 41 (9 self)
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Abstract. We study the proof–theoretic strength and effective content denote Ramof the infinite form of Ramsey’s theorem for pairs. Let RT n k sey’s theorem for k–colorings of n–element sets, and let RT n < ∞ denote (∀k)RTn k. Our main result on computability is: For any n ≥ 2 and any computable (recursive) k–coloring of the n–element sets of natural numbers, there is an infinite homogeneous set X with X ′ ′ ≤T 0 (n). Let I�n and B�n denote the �n induction and bounding schemes, respectively. Adapting the case n = 2 of the above result (where X is low2) to models is conservative of arithmetic enables us to show that RCA0 + I �2 + RT2 2 over RCA0 + I �2 for �1 1 statements and that RCA0 + I �3 + RT2 < ∞ is �1 1conservative over RCA0 + I �3. It follows that RCA0 + RT2 2 does not imply B �3. In contrast, J. Hirst showed that RCA0 + RT2 < ∞ does imply B �3, and we include a proof of a slightly strengthened version of this result. It follows that RT2 < ∞ is strictly stronger than RT2 2 over RC A0. 1.
Higher Order Logic
 In Handbook of Logic in Artificial Intelligence and Logic Programming
, 1994
"... Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Definin ..."
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Cited by 19 (0 self)
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Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Defining data types : : : : : : : : : : : : : : : : : : : : : 6 2.4 Describing processes : : : : : : : : : : : : : : : : : : : : : 8 2.5 Expressing convergence using second order validity : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.6 Truth definitions: the analytical hierarchy : : : : : : : : 10 2.7 Inductive definitions : : : : : : : : : : : : : : : : : : : : : 13 3 Canonical semantics of higher order logic : : : : : : : : : : : : 15 3.1 Tarskian semantics of second order logic : : : : : : : : : 15 3.2 Function and re
Elimination of Skolem functions for monotone formulas in analysis
 ARCHIVE FOR MATHEMATICAL LOGIC
"... ..."
On the Strength of Ramsey's Theorem
 Notre Dame J. Formal Logic
, 1995
"... this paper we study the logical strength of Ramsey's Theorem (1930), especially of Ramsey's Theorem for partitions of pairs into two pieces. ..."
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Cited by 12 (0 self)
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this paper we study the logical strength of Ramsey's Theorem (1930), especially of Ramsey's Theorem for partitions of pairs into two pieces.
The Baire category theorem in weak subsystems of secondorder arithmetic
 THE JOURNAL OF SYMBOLIC LOGIC
, 1993
"... ..."
ComputabilityTheoretic and ProofTheoretic Aspects of Partial and Linear Orderings
 Israel Journal of mathematics
"... Szpilrajn's Theorem states that any partial order P = hS;
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Cited by 9 (0 self)
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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.
Π 0 1 sets and models of WKL0
"... We show that any two Medvedev complete Π 0 1 subsets of 2 ω are recursively homeomorphic. We obtain a Π 0 1 set Q ′ of countable coded ωmodels of WKL0 with a strong homogeneity property. We show that if G is a generic element of Q ′ , then the ωmodel of WKL0 coded by G satisfies ∀X∀Y (if X is de ..."
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Cited by 8 (5 self)
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We show that any two Medvedev complete Π 0 1 subsets of 2 ω are recursively homeomorphic. We obtain a Π 0 1 set Q ′ of countable coded ωmodels of WKL0 with a strong homogeneity property. We show that if G is a generic element of Q ′ , then the ωmodel of WKL0 coded by G satisfies ∀X∀Y (if X is definable from Y, then X is Turing reducible to Y). We use a result of Kučera to refute some plausible conjectures concerning ωmodels of WKL0. We generalize our results to nonωmodels of WKL0. We discuss the significance of our results for foundations of mathematics.
A.: How much incomputable is the separable HahnBanach Theorem
 Conference on Computability and Complexity in Analysis. Number 348 in Informatik Berichte, FernUniversität Hagen (2008) 101 – 117
"... Abstract. We determine the computational complexity of the HahnBanach Extension Theorem. To do so, we investigate some basic connections between reverse mathematics and computable analysis. In particular, we use Weak König’s Lemma within the framework of computable analysis to classify incomputable ..."
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Cited by 7 (2 self)
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Abstract. We determine the computational complexity of the HahnBanach Extension Theorem. To do so, we investigate some basic connections between reverse mathematics and computable analysis. In particular, we use Weak König’s Lemma within the framework of computable analysis to classify incomputable functions of low complexity. By defining the multivalued function Sep and a natural notion of reducibility for multivalued functions, we obtain a computational counterpart of the subsystem of second order arithmetic WKL0. We study analogies and differences between WKL0 and the class of Sepcomputable multivalued functions. Extending work of Brattka, we show that a natural multivalued function associated with the HahnBanach Extension Theorem is Sepcomplete. 1.
Reverse Mathematics and Recursive Graph Theory
 Math. Log. Quart
, 1998
"... Abstract. We examine a number of results of infinite combinatorics using the techniques of reverse mathematics. Our results are inspired by similar results in recursive combinatorics. Theorems included concern colorings of graphs and bounded graphs, Euler paths, and Hamilton paths. Reverse mathemati ..."
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Cited by 4 (1 self)
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Abstract. We examine a number of results of infinite combinatorics using the techniques of reverse mathematics. Our results are inspired by similar results in recursive combinatorics. Theorems included concern colorings of graphs and bounded graphs, Euler paths, and Hamilton paths. Reverse mathematics provides powerful techniques for analyzing the logical content of theorems. By contrast, recursive mathematics analyzes the effective content of theorems. In many cases, theorems of reverse mathematics have recursion theoretic corollaries. Conversely, theorems and techniques of recursive mathematics can often inspire related results in reverse mathematics, as demonstrated by the research presented here. In Section 1, a brief description of reverse mathematics is given. Sections 2 and 3 analyze theorems on graph colorings. Section 4 considers graphs with Euler paths. Stronger axiom systems are introduced in Section 5 and applied to the study of Hamilton paths in Section 6. 1. Reverse mathematics. In [4], Friedman defined subsystems of secondorder arithmetic useful in determining the prooftheoretic and recursiontheoretic strength of theorems. The language of