Reverse mathematics and order theoretic fixed point theorems

Takashi Sato, Takeshi Yamazaki

Research output: Contribution to journalArticlepeer-review

Abstract

The theory of countable partially ordered sets (posets) is developed within a weak subsystem of second order arithmetic. We within RCA0 give definitions of notions of the countable order theory and present some statements of countable lattices equivalent to arithmetical comprehension axiom over RCA0. Then we within RCA0 give proofs of Knaster–Tarski fixed point theorem, Tarski–Kantorovitch fixed point theorem, Bourbaki–Witt fixed point theorem, and Abian–Brown maximal fixed point theorem for countable lattices or posets. We also give Reverse Mathematics results of the fixed point theory of countable posets; Abian–Brown least fixed point theorem, Davis’ converse for countable lattices, Markowski’s converse for countable posets, and arithmetical comprehension axiom are pairwise equivalent over RCA0. Here the converses state that some fixed point properties characterize the completeness of the underlying spaces.

Original languageEnglish
Pages (from-to)385-396
Number of pages12
JournalArchive for Mathematical Logic
Volume56
Issue number3-4
DOIs
Publication statusPublished - 2017 May 1

Keywords

  • Countable lattices
  • Countable posets
  • Fixed point theorem
  • Order theory
  • Reverse mathematics
  • Second order arithmetic

ASJC Scopus subject areas

  • Philosophy
  • Logic

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