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.

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.

scholarly journals Brouwer’s Weak Counterexamples and the Creative Subject: A Critical Survey

2020 ◽  
Vol 49 (6) ◽  
pp. 1111-1157
Author(s):  
Peter Fletcher
Keyword(s):  
Mathematical Work ◽  
Weak Continuity ◽  
Critical Survey ◽  
Continuity Theorem ◽  
Continuity Principle ◽  
Mathematical Proofs ◽  
Choice Sequence ◽  
Creative Subject

AbstractI survey Brouwer’s weak counterexamples to classical theorems, with a view to discovering (i) what useful mathematical work is done by weak counterexamples; (ii) whether they are rigorous mathematical proofs or just plausibility arguments; (iii) the role of Brouwer’s notion of the creative subject in them, and whether the creative subject is really necessary for them; (iv) what axioms for the creative subject are needed; (v) what relation there is between these arguments and Brouwer’s theory of choice sequences. I refute one of Brouwer’s claims with a weak counterexample of my own. I also examine Brouwer’s 1927 proof of the negative continuity theorem, which appears to be a weak counterexample reliant on both the creative subject and the concept of choice sequence; I argue that it provides a good justification for the weak continuity principle, but it is not a weak counterexample and it does not depend essentially on the creative subject.

Sign in / Sign up

Export Citation Format

Share Document