TY - JOUR

T1 - Σ-Preorderings in ℍF (ℝ)

AU - Morozov, A. S.

PY - 2019/11/1

Y1 - 2019/11/1

N2 - 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.

AB - 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.

KW - Hereditarily finite superstructure

KW - Ordinal

KW - Real numbers

KW - Σ-definable preordering

KW - Sigma-definable preordering

KW - DEFINABILITY

KW - NONPRESENTABILITY

UR - http://www.scopus.com/inward/record.url?scp=85076336469&partnerID=8YFLogxK

U2 - 10.1007/s10469-019-09560-0

DO - 10.1007/s10469-019-09560-0

M3 - Article

AN - SCOPUS:85076336469

VL - 58

SP - 405

EP - 416

JO - Algebra and Logic

JF - Algebra and Logic

SN - 0002-5232

IS - 5

ER -