Results 1  10
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11
Partial realizations of Hilbert’s program
 Journal of Symbolic Logic
, 1988
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Cited by 38 (8 self)
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JSTOR is a notforprofit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Association for Symbolic Logic is collaborating with JSTOR to digitize, preserve and extend access to The
Located Sets And Reverse Mathematics
 Journal of Symbolic Logic
, 1999
"... Let X be a compact metric space. A closed set K is located if the distance function d(x, K) exists as a continuous realvalued function on X ; weakly located if the predicate d(x, K) > r is # 1 allowing parameters. The purpose of this paper is to explore the concepts of located and weakly loca ..."
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Cited by 13 (5 self)
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Let X be a compact metric space. A closed set K is located if the distance function d(x, K) exists as a continuous realvalued function on X ; weakly located if the predicate d(x, K) > r is # 1 allowing parameters. The purpose of this paper is to explore the concepts of located and weakly located subsets of a compact separable metric space in the context of subsystems of second order arithmetic such as RCA 0 , WKL 0 and ACA 0 . We also give some applications of these concepts by discussing some versions of the Tietze extension theorem. In particular we prove an RCA 0 version of this result for weakly located closed sets.
The Baire category theorem in weak subsystems of secondorder arithmetic
 THE JOURNAL OF SYMBOLIC LOGIC
, 1993
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Separable Banach space theory needs strong set existence axioms
 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
, 1996
"... We investigate the strength of set existence axioms needed for separable Banach space theory. We show that a very strong axiom, Π1 1 comprehension, is needed to prove such basic facts as the existence of the weak∗ closure of any normclosed subspace of ℓ1 = c ∗ 0. This is in contrast to earlier w ..."
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Cited by 7 (5 self)
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We investigate the strength of set existence axioms needed for separable Banach space theory. We show that a very strong axiom, Π1 1 comprehension, is needed to prove such basic facts as the existence of the weak∗ closure of any normclosed subspace of ℓ1 = c ∗ 0. This is in contrast to earlier work [6, 4, 7, 23, 22] in which theorems of separable Banach space theory were proved in very weak subsystems of second order arithmetic, subsystems which are conservative over Primitive Recursive Arithmetic for Π0 2 sentences. En route to our main results, we prove the Krein ˇ Smulian theorem in ACA0, and we give a new, elementary proof of a result of McGehee on weak ∗ sequential closure ordinals.
Separation and weak König’s lemma
 Journal of Symbolic Logic
, 1999
"... investigating the strength of set existence axioms needed for separable Banach space theory. We show that the separation theorem foropenconvexsetsisequivalenttoWKL0 over RCA0. Weshow that the separation theorem for separably closed convex sets is equivalent to ACA0 over RCA0. Our strategy for provin ..."
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Cited by 5 (2 self)
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investigating the strength of set existence axioms needed for separable Banach space theory. We show that the separation theorem foropenconvexsetsisequivalenttoWKL0 over RCA0. Weshow that the separation theorem for separably closed convex sets is equivalent to ACA0 over RCA0. Our strategy for proving these geometrical Hahn–Banach theorems is to reduce to the finitedimensional case by means of a compactness argument. 1.
On the logical analysis of proofs based on nonseparable Hilbert space theory
, 2010
"... Starting in [15] and then continued in [9, 17, 24] and [18], general logical metatheorems were developed that guarantee the extractability of highly uniform effective bounds from proofs of theorems that hold for general classes of structures such as ..."
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Cited by 4 (4 self)
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Starting in [15] and then continued in [9, 17, 24] and [18], general logical metatheorems were developed that guarantee the extractability of highly uniform effective bounds from proofs of theorems that hold for general classes of structures such as
Lebesgue numbers and Atsuji spaces in subsystems of secondorder arithmetic
 Arch. Math. Logic
, 1998
"... Abstract. We study Lebesgue and Atsuji spaces within subsystems of second order arithmetic. The former spaces are those such that every open covering has a Lebesgue number, while the latter are those such that every continuous function defined on them is uniformly continuous. The main results we obt ..."
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Cited by 4 (1 self)
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Abstract. We study Lebesgue and Atsuji spaces within subsystems of second order arithmetic. The former spaces are those such that every open covering has a Lebesgue number, while the latter are those such that every continuous function defined on them is uniformly continuous. The main results we obtain are the following: the statement “every compact space is Lebesgue ” is equivalent to WKL0; the statements “every perfect Lebesgue space is compact ” and “every perfect Atsuji space is compact ” are equivalent to ACA0; the statement “every Lebesgue space is Atsuji ” is provable in RCA0; the statement “every Atsuji space is Lebesgue ” is provable in ACA0. We also prove that the statement “the distance from a closed set is a continuous function ” is equivalent to Π 1 1CA0.
Vitali’s theorem and WWKL
 Archive for Mathematical Logic
"... Abstract. Continuing the investigations of X. Yu and others, we study the role of set existence axioms in classical Lebesgue measure theory. We show that pairwise disjoint countable additivity for open sets of reals is provable in RCA0. We show that several wellknown measuretheoretic propositions ..."
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Cited by 2 (0 self)
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Abstract. Continuing the investigations of X. Yu and others, we study the role of set existence axioms in classical Lebesgue measure theory. We show that pairwise disjoint countable additivity for open sets of reals is provable in RCA0. We show that several wellknown measuretheoretic propositions including the Vitali Covering Theorem are equivalent to WWKL over RCA0. 1.
On a Question of Brown and Simpson
, 1994
"... this paper, we introduce an equivalent formulation of BCTII , which we denote BCT# ..."
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Cited by 1 (0 self)
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this paper, we introduce an equivalent formulation of BCTII , which we denote BCT#
Gödel functional interpretation and weak compactness
, 2011
"... In recent years, proof theoretic transformations (socalled proof interpretations) that are based on extensions of monotone forms of Gödel’s famous functional (‘Dialectica’) interpretation have been used systematically to extract new content from proofs in abstract nonlinear analysis. This content c ..."
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Cited by 1 (1 self)
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In recent years, proof theoretic transformations (socalled proof interpretations) that are based on extensions of monotone forms of Gödel’s famous functional (‘Dialectica’) interpretation have been used systematically to extract new content from proofs in abstract nonlinear analysis. This content consists both in effective quantitative bounds as well as in qualitative uniformity results. One of the main ineffective tools in abstract functional analysis is the use of sequential forms of weak compactness. As we recently verified, the sequential form of weak compactness for bounded closed and convex subsets of an abstract (not necessarily separable) Hilbert space can be carried out in suitable formal systems that are covered by existing metatheorems developed in the course of the proof mining program. In particular, it follows that the monotone functional interpretation of this weak compactness principle can be realized by a functional Ω ∗ definable from bar recursion (in the sense of Spector) of lowest type. While a case study on the analysis of strong convergence results (due to Browder and Wittmann resp.) that are based on weak compactness indicates that the use of the latter seems to be eliminable, things apparently are different for weak convergence theorems such as the famous Baillon nonlinear ergodic theorem. For this theorem we recently extracted an