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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.
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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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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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
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 ..."
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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.
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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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
Principles Weaker than BDN
, 2012
"... BDN is a weak principle of constructive analysis. Several interesting principles implied by BDN have already been identified, namely the closure of the antiSpecker spaces under product, the Riemann Permutation Theorem, and the Cauchyness of all partially Cauchy sequences. Here these are shown to ..."
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BDN is a weak principle of constructive analysis. Several interesting principles implied by BDN have already been identified, namely the closure of the antiSpecker spaces under product, the Riemann Permutation Theorem, and the Cauchyness of all partially Cauchy sequences. Here these are shown to be strictly weaker than BDN, yet not provable in set theory alone under constructive logic.
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 ..."
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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 Constructive Study of Landau’s Summability Theorem
"... Abstract: A summability theorem of Landau, which classically is a simple consequence of the uniform boundedness theorem, is examined within Bishopstyle constructive mathematics. It is shown that the original theorem is nonconstructive, and that a natural weakening of the theorem is constructively ..."
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Abstract: A summability theorem of Landau, which classically is a simple consequence of the uniform boundedness theorem, is examined within Bishopstyle constructive mathematics. It is shown that the original theorem is nonconstructive, and that a natural weakening of the theorem is constructively equivalent to Ishihara’s principle BDN. The paper ends with a number of results that, while not as strong as Landau’s theorem, nevertheless contain positive computational information related to its conclusion. Key Words: constructive, lp space, summability, Landau
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.
On Choice Principles and Fan Theorems
"... Abstract: Veldman proved that the contrapositive of countable binary choice is a theorem of fullfledged intuitionism, to which end he used a principle of continuous choice and the fan theorem. It has turned out that continuous choice is unnecessary in this context, and that a weak form of the fan t ..."
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Abstract: Veldman proved that the contrapositive of countable binary choice is a theorem of fullfledged intuitionism, to which end he used a principle of continuous choice and the fan theorem. It has turned out that continuous choice is unnecessary in this context, and that a weak form of the fan theorem suffices which holds in the presence of countable choice. In particular, the contrapositive of countable binary choice is valid in Bishopstyle constructive mathematics. We further discuss a generalisation of this result and link it to Ishihara’s boundedness principle BDN. Key Words: constructive mathematics, fan theorem, countable choice