The Monotone Completeness Theorem in Constructive Reverse Mathematics

2019 ◽  
pp. 101-112
Author(s):  
Hajime Ishihara ◽  
Takako Nemoto
Keyword(s):  
Completeness Theorem ◽  
Reverse Mathematics ◽  
Constructive Reverse Mathematics

Variations on a theme by Ishihara

2014 ◽  
Vol 25 (7) ◽  
pp. 1569-1577 ◽  
Author(s):  
HANNES DIENER
Keyword(s):  
Short Note ◽  
Classification Problem ◽  
Reverse Mathematics ◽  
Baire Space ◽  
Partial Answer ◽  
Constructive Mathematics ◽  
Natural Numbers ◽  
Variations On A Theme ◽  
Constructive Reverse Mathematics ◽  
Rule Out

Ishihara's tricks have proven to be a highly useful tool in constructive mathematics, since they enable one to make decisions that seem, on first glance, impossible. They do, however, require that one deals with strongly extensional mappings on complete spaces. In this short note, we show how these assumptions can be weakened. Furthermore, we apply these generalizations to give a partial answer to the question, whether constructively we can rule out the existence of injections from Baire space into the natural numbers, to a version of Riemann's per mutation theorem and to a classification problem about cardinalities in constructive reverse mathematics.

A continuity principle, a version of Baire's theorem and a boundedness principle

2008 ◽  
Vol 73 (4) ◽  
pp. 1354-1360 ◽  
Author(s):  
Hajime Ishihara ◽  
Peter Schuster
Keyword(s):  
Reverse Mathematics ◽  
Weak Continuity ◽  
Baire’S Theorem ◽  
Continuity Principle ◽  
Restricted Form ◽  
Constructive Reverse Mathematics

AbstractWe deal with a restricted form WC-N′ of the weak continuity principle, a version BT′ of Baire's theorem, and a boundedness principle BD-N. We show, in the spirit of constructive reverse mathematics, that WC-N′, BT′ + ¬LPO and BD-N + ¬LPO are equivalent in a constructive system, where LPO is the limited principle of omniscience.

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