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.
- Hereditarily finite superstructure
- Real numbers
- Σ-definable preordering
- Sigma-definable preordering