Σ-Preorderings in ℍF (ℝ)

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Abstract

It is proved that the ordinal ω1cannot be embedded into a preordering Σ-definable with parameters in the hereditarily finite superstructure over the real numbers. As a corollary, we obtain the descriptions of ordinals Σ-presentable overℍF(ℝ) and of Gödel constructive sets of the form Lα. It is also shown that there are no Σ-presentations of structures of T-, m-, 1- and tt-degrees.

Original languageEnglish
Pages (from-to)405-416
Number of pages12
JournalAlgebra and Logic
Volume58
Issue number5
DOIs
Publication statusPublished - 1 Nov 2019

Keywords

  • Hereditarily finite superstructure
  • Ordinal
  • Real numbers
  • Σ-definable preordering
  • Sigma-definable preordering
  • DEFINABILITY
  • NONPRESENTABILITY

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