On principles between Σ1 - And Σ2 -induction, and monotone enumerations

Alexander P. Kreuzer, Keita Yokoyama

Research output: Contribution to journalArticlepeer-review

8 Citations (Scopus)


We show that many principles of first-order arithmetic, previously only known to lie strictly between Σ1-induction and Σ2-induction, are equivalent to the well-foundedness of ωω. Among these principles are the iteration of partial functions (PΣ1) of Hájek and Paris, the bounded monotone enumerations principle (non-iterated, BME1) by Chong, Slaman, and Yang, the relativized Paris-Harrington principle for pairs, and the totality of the relativized Ackermann-Péter function. With this we show that the well-foundedness of ωω is a far more widespread than usually suspected. Further, we investigate the k-iterated version of the bounded monotone iterations principle (BMEk), and show that it is equivalent to the well-foundedness of the (k + 1)-height ω-tower ωω.

Original languageEnglish
Article number1650004
JournalJournal of Mathematical Logic
Issue number1
Publication statusPublished - 2016 Jun 1
Externally publishedYes


  • Ackermann function
  • bounded monotone enumerations
  • Fragments of arithmetics
  • ordinal numbers
  • Paris-Harrington theorem
  • reverse mathematics

ASJC Scopus subject areas

  • Logic


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