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On Weak Markov's Principle
Author(s) -
Kohlenbach Ulrich
Publication year - 2002
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/1521-3870(200210)48:1+<59::aid-malq59>3.0.co;2-i
Subject(s) - mathematics , constructive , schema (genetic algorithms) , law of excluded middle , markov chain , comprehension , calculus (dental) , pure mathematics , algebra over a field , discrete mathematics , epistemology , computer science , linguistics , statistics , philosophy , machine learning , programming language , process (computing) , medicine , dentistry
We show that the so‐called weak Markov's principle (WMP) which states that every pseudo‐positive real number is positive is underivable in ω ≔ E‐HA ω + AC. Since ω allows one to formalize (atl eastl arge parts of) Bishop 's constructive mathematics, this makes it unlikely that WMP can be proved within the framework of Bishop ‐style 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 one to derive the law of excluded middle for such formulas.