Σ-Preorderings in \( \mathbb{H}\mathbbm{F} \)(ℝ)

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 over\( \mathbb{H}\mathbbm{F} \)(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.