The Shelah HOD(S) model and its real-ordinal presentation

Work in the generic extension $V[G]$ of the CH-length Shelah construction Shelah's CH-length homogeneous sweet construction started over the constructible universe $L$. Let $S={\bigcup }_{\alpha \in \mathrm{O}\mathrm{r}\mathrm{d}}{}^{\omega }\alpha$ be the class of all countable sequences of ordinals, exactly as in the Solovay $HOD(S)$ presentation The hereditarily ordinal-sequence-definable Solovay model, and define

$N=HOD(S)=\{x:\mathrm{tc}(\{x\})\subseteq OD(S)\},$

where $OD(S)$ is the class of sets uniquely definable in a rank from one $s\in S$, finitely many ordinals and a formula, in the sense of Ordinal definability and HOD. The definition is the same uniform first-order class as in the Solovay presentation, with the ambient model now being the Shelah extension rather than a Lévy collapse.

Conventions proved in this pair. Finite and countable tuples of members of $S$ interleave into one member of $S$ by a fixed pairing function on $\omega$, so "one $S$-parameter" loses no generality; a real, viewed as a binary sequence of ordinals, is itself a member of $S$; and every real belongs to $N$, because a real is definable from itself as an $S$-parameter.

Real-ordinal presentation. In this branch the ambient ground model is $L$. By the countably-generated-support coding of Real names are captured and coded meagre unions are absorbed, every set $A\subseteq \mathbb{R}$ in $N$ is definable from one real together with finitely many ordinals: take the $B$-name of the $S$-parameter defining $A$, capture its countably many deciding antichains in a stage $B_{\alpha }$, record the generic's chosen index in each of them by a single real, and code the ground-model name and the canonical enumerations by ordinals, which lie in $L$. Conversely, a real together with finitely many ordinals interleaves into one member of $S$. This is the real-and-ordinal presentation required by the equiconsistency statement; it is asserted only in the branch over $L$ and is not claimed for an arbitrary ground extension.

No equality with $L(\mathbb{R})$, with $HOD(\mathbb{R})$, or with any class built from all ${\omega }_1$-sequences of ordinals is asserted. The ambient ${\omega }_1$ is not collapsed: the construction is ccc and adds no new ordinals, so the ordinal height of $N$ is that of $V[G]$.