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Sequential continuity of linear mappings in constructive mathematics
 J. Universal Computer Science
, 1997
"... Abstract: This paper deals, constructively, with two theorems on the sequential continuity of linear mappings. The classical proofs of these theorems use the boundedness of the linear mappings, which is a constructively stronger property than sequential continuity; and constructively inadmissable ve ..."
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Abstract: This paper deals, constructively, with two theorems on the sequential continuity of linear mappings. The classical proofs of these theorems use the boundedness of the linear mappings, which is a constructively stronger property than sequential continuity; and constructively inadmissable versions of the BanachSteinhaus theorem.
On weak Markov's principle
 MLQ MATH. LOG. Q
, 2002
"... We show that the socalled weak Markov's principle (WMP) which states that every pseudopositive real number is positive is underivable in T # :=EHA # +AC. Since T # allows to formalize (at least large parts of) Bishop's constructive mathematics this makes it unlikely that WMP can be proved ..."
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Cited by 4 (1 self)
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We show that the socalled weak Markov's principle (WMP) which states that every pseudopositive real number is positive is underivable in T # :=EHA # +AC. Since T # allows to formalize (at least large parts of) Bishop's constructive mathematics this makes it unlikely that WMP can be proved within the framework of Bishopstyle mathematics (which has been open for about 20 years). The underivability even holds if the ine#ective schema of full comprehension (in all types) for negated formulas (in particular for #free formulas) is added which allows to derive the law of excluded middle for such formulas.
Intuitionistic notions of boundedness in N
, 2008
"... We consider notions of boundedness of subsets of the natural numbers N that occur when doing mathematics in the context of intuitionistic logic. We obtain a new characterization of the notion of a pseudobounded subset and formulate the closely related notion of a detachably finite subset. We establi ..."
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Cited by 4 (0 self)
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We consider notions of boundedness of subsets of the natural numbers N that occur when doing mathematics in the context of intuitionistic logic. We obtain a new characterization of the notion of a pseudobounded subset and formulate the closely related notion of a detachably finite subset. We establish metric equivalents for a subset of N to be detachably finite and to satisfy the ascending chain condition. Following Ishihara, we spell out the relationship between detachable finiteness and sequential continuity. Most of the results do not require countable choice.
Sequentially Continuous Linear Mappings in Constructive Analysis
, 1996
"... this paper we derive some results about sequentially continuous linear mappings within BISH. These results tend to reinforce our hope that such mappings may turn out to be bounded (continuous) after all. For background material on BISH, see [1], and for information about the relation between BISH, I ..."
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Cited by 3 (3 self)
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this paper we derive some results about sequentially continuous linear mappings within BISH. These results tend to reinforce our hope that such mappings may turn out to be bounded (continuous) after all. For background material on BISH, see [1], and for information about the relation between BISH, INT, and RUSS, see [2]. 2 Sequential continuity preserves Cauchyness
Canonical Effective Subalgebras of Classical Algebras as Constructive Metric Completions
"... Abstract. We prove general theorems about unique existence of effective subalgebras of classical algebras. The theorems are consequences of standard facts about completions of metric spaces within the framework of constructive mathematics, suitably interpreted in realizability models. We work with g ..."
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Cited by 2 (0 self)
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Abstract. We prove general theorems about unique existence of effective subalgebras of classical algebras. The theorems are consequences of standard facts about completions of metric spaces within the framework of constructive mathematics, suitably interpreted in realizability models. We work with general realizability models rather than with a particular model of computation. Consequently, all the results are applicable in various established schools of computability, such as type 1 and type 2 effectivity, domain representations, equilogical spaces, and others. 1
A Constructive Study of Landau's . . .
, 2009
"... A summability theorem of Landau, which classically is a simple consequence of the uniform boundedness theorem, is examined constructively. ..."
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A summability theorem of Landau, which classically is a simple consequence of the uniform boundedness theorem, is examined constructively.
Realizability Models Refuting Ishihara’s Boundedness Principle
, 2011
"... In [Ish92] H. Ishihara introduced the socalled boundedness principle BDN which claims that every countable pseudobounded subset of N is bounded. Here S ⊆ N is called pseudobounded iff for every sequence a ∈ SN there exists an n ∈ N such that ak < k for all k ≥ n. 1 Obviously, the principle BDN is ..."
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In [Ish92] H. Ishihara introduced the socalled boundedness principle BDN which claims that every countable pseudobounded subset of N is bounded. Here S ⊆ N is called pseudobounded iff for every sequence a ∈ SN there exists an n ∈ N such that ak < k for all k ≥ n. 1 Obviously, the principle BDN is
A note on Brouwer’s weak continuity principle and the transfer principle in nonstandard analysis
, 2011
"... A wellknown model of nonstandard analysis is obtained by extending the structure of real numbers using an ultra power construction. A constructive approach due to Schmieden and Laugwitz uses instead a reduced power construction modulo a cofinite filter, but has the drawback that the transfer princi ..."
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A wellknown model of nonstandard analysis is obtained by extending the structure of real numbers using an ultra power construction. A constructive approach due to Schmieden and Laugwitz uses instead a reduced power construction modulo a cofinite filter, but has the drawback that the transfer principle is weak. In this paper it is shown that this principle can be strengthened by employing Brouwerian continuity axioms familiar from intuitionistic systems. We end by commenting on the relation between the transfer principle and Ishihara’s boundedness principle. 1
Documenta Math. 973 Inheriting the AntiSpecker Property
, 2009
"... Abstract. The antithesis of Specker’s theorem from recursive analysis is further examined from Bishop’s constructive viewpoint, with particular attention to its passage to subspaces and products. Ishihara’s principle BDN comes into play in the discussion of products with the antiSpecker property. ..."
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Abstract. The antithesis of Specker’s theorem from recursive analysis is further examined from Bishop’s constructive viewpoint, with particular attention to its passage to subspaces and products. Ishihara’s principle BDN comes into play in the discussion of products with the antiSpecker property.
Realizability Models Refuting Ishihara’s Boundedness Principle
"... Ishihara’s Boundedness Principle BDN was introduced in [Ish92] and has turned out to be most useful for constructive analysis, see e.g. [Ish01]. It is equivalent to the statement that every sequentially continuous function from NN to N is continuous w.r.t. the usual metric topology on NN. We constr ..."
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Ishihara’s Boundedness Principle BDN was introduced in [Ish92] and has turned out to be most useful for constructive analysis, see e.g. [Ish01]. It is equivalent to the statement that every sequentially continuous function from NN to N is continuous w.r.t. the usual metric topology on NN. We construct models for higher order arithmetic and intuitionistic set theory in which both every function from N N to N is sequentially continuous and in which the axiom of choice from N N to N holds. Since the latter is known to be inconsistent with the statement that all functions from N N to N are continuous these models refute BDN.