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14
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 l ..."
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Cited by 15 (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
"... ..."
Ordered Groups: A Case Study In Reverse Mathematics
 Bulletin of Symbolic Logic
, 1999
"... this article, we will be concerned only with fully ordered groups and will use the term ordered group to mean fully ordered group. There are a number of group conditions which imply full orderability. The simplest is given by the following classical theorem. ..."
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Cited by 9 (2 self)
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this article, we will be concerned only with fully ordered groups and will use the term ordered group to mean fully ordered group. There are a number of group conditions which imply full orderability. The simplest is given by the following classical theorem.
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.
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.
The HahnBanach Theorem in Type Theory
, 1997
"... We give the basic deønitions for pointfree functional analysis and present constructive proofs of the Alaoglu and HahnBanach theorems in the setting of formal topology. 1 Introduction We present the basic concepts and deønitions needed in a pointfree approach to functional analysis via formal topol ..."
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Cited by 7 (0 self)
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We give the basic deønitions for pointfree functional analysis and present constructive proofs of the Alaoglu and HahnBanach theorems in the setting of formal topology. 1 Introduction We present the basic concepts and deønitions needed in a pointfree approach to functional analysis via formal topology. Our main results are the constructive proofs of localic formulations of the Alaoglu and HellyHahnBanach 1 theorems. Earlier pointfree formulations of the HahnBanach theorem, in a topostheoretic setting, were presented by Mulvey and Pelletier (1987,1991) and by Vermeulen (1986). A constructive proof based on points was given by Bishop (1967). In the formulation of his proof, the norm of the linear functional is preserved to an arbitrary degree by the extension and a counterexample shows that the norm, in general, is not preserved exactly. As usual in pointfree topology, our guideline is to deøne the objects under analysis as formal points of a suitable formal space. After this has...
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 6 (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.
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 5 (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.