**February 21**

**Andrey Morozov**,
Novosibirsk State University

**On $\Sigma$-preorderings in HF(R)**

We prove that $\omega_1$ cannot be embedded into any preordering $\Sigma$-definable with parameters in the hereditarily finite superstructure over the ordered field of real numbers, HF(R). As corollaries, we obtain characterizations of $\Sigma$-presentable ordinals and GĂ¶del constructive sets of kind $L_\alpha$. It also follows that there are no $\Sigma$-presentations for structures of $T$-, $m$-, $1$-, and $tt$-degrees over HF(R).