Open AccessOn Weak Markov's Principle
Author(s) -
Ulrich Kohlenbach
Publication year - 2001
Publication title -
brics report series
Language(s) - English
Resource type - Journals
eISSN - 1601-5355
pISSN - 0909-0878
DOI - 10.7146/brics.v8i51.21712
Subject(s) - omega , constructive , mathematics , law of excluded middle , schema (genetic algorithms) , markov chain , calculus (dental) , markov process , discrete mathematics , computer science , epistemology , linguistics , philosophy , statistics , machine learning , programming language , medicine , dentistry , process (computing)
We show that the so-called weak Markov's principle (WMP) which states that every pseudo-positive real number is positive is underivable in T^{omega}:= E-HA^{omega} + AC. Since T^{omega} 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 Bishop-style mathematics (which has been open for about 20 years). The underivability even holds if the ineffective schema of full comprehension (in all types) for negated formulas (in particular for $\exists$-free formulas) is added which allows to derive the law of excluded middle for such formulas.