$\Sigma$-preorderings in ${\mathbb{HF}(\mathbb{R})}$

It is proved that the ordinal $\omega_1$ cannot be embedded into a preordering $\Sigma$-definable with parameters in the hereditarily finite superstructure over the real numbers. As a corollary, we obtain the descriptions of ordinals $\Sigma$-presentable over ${\mathbb{HF}(\mathbb{R})}$ and of Gödel constructive sets of the form $L_\alpha$. It is also shown that there are no $\Sigma$-presentations of structures of $T$-, $m$-, $1$- and $tt$-degrees.