Mathematische Semesterberichte

, Volume 57, Issue 1, pp 57–72 | Cite as

Kronecker’s density theorem and irrational numbers in constructive reverse mathematics

  • Hajime Ishihara
  • Peter Schuster
Mathematik in der Forschung


To prove Kronecker’s density theorem in Bishop-style constructive analysis one needs to define an irrational number as a real number that is bounded away from each rational number. In fact, once one understands “irrational” merely as “not rational”, then the theorem becomes equivalent to Markov’s principle. To see this we undertake a systematic classification, in the vein of constructive reverse mathematics, of logical combinations of “rational” and “irrational” as predicates of real numbers.

Mathematics subject classification (2000)

03F60 11J71 11J72 


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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.School of Information ScienceJapan Advanced Institute of Science and TechnologyIshikawaJapan
  2. 2.Department of Pure MathematicsUniversity of LeedsLeedsUK

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