Results 1 
3 of
3
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 ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
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.
Constructive Order Completeness
, 2004
"... Partially ordered sets are investigated from the point of view of Bishop’s constructive mathematics. Unlike the classical case, one cannot prove constructively that every nonempty bounded above set of real numbers has a supremum. However, the order completeness of R is expressed constructively by an ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
Partially ordered sets are investigated from the point of view of Bishop’s constructive mathematics. Unlike the classical case, one cannot prove constructively that every nonempty bounded above set of real numbers has a supremum. However, the order completeness of R is expressed constructively by an equivalent condition for the existence of the supremum, a condition of (upper) order locatedness which is vacuously true in the classical case. A generalization of this condition will provide a definition of upper locatedness for a partially ordered set. It turns out that the supremum of a set S exists if and only if S is upper located and has a weak supremum—that is, the classical least upper bound. A partially ordered set will be called order complete if each nonempty subset that is bounded above and upper located has a supremum. It can be proved that, as in the classical mathematics, R n is order complete. 1
ANOTHER UNIQUE WEAK KÖNIG’S LEMMA WKL!!
"... Abstract. In [2] J. Berger and Ishihara proved, via a circle of informal implications involving countable choice, that Brouwer’s Fan Theorem for detachable bars on the binary fan is equivalent in Bishop’s sense to various principles including a version WKL! of Weak König’s Lemma with a strong effect ..."
Abstract
 Add to MetaCart
Abstract. In [2] J. Berger and Ishihara proved, via a circle of informal implications involving countable choice, that Brouwer’s Fan Theorem for detachable bars on the binary fan is equivalent in Bishop’s sense to various principles including a version WKL! of Weak König’s Lemma with a strong effective uniqueness hypothesis. Schwichtenberg [9] proved the equivalence directly and formalized his proof in Minlog. We verify that his result does not require countable choice, and derive a separation principle SP from the Fan Theorem, in a minimal intuitionistic system M of analysis with function comprehension. In contrast, WKL!! comes from Weak König’s lemma WKL by adding the hypothesis that any two infinite paths must agree. WKL!! is interderivable over M with the conjunction of a consequence of Markov’s Principle and the double negation of WKL. This decomposition is in the spirit of Ishihara’s [4] and J. Berger’s [1]. Kleene’s function realizability and the author’s modified realizability establish that WKL!! is strictly weaker than WKL and strictly stronger than WKL!.