Constructibility real degrees in the side-by-side Sacks model

We study the join-semilattice of constructibility real degrees in the side-by-side Sacks model, the model of set theory obtained by forcing with a countable-support product of infinitely many Sacks forcings over the constructible universe. In particular, we prove that in the side-by-side Sacks model the join-semilattice of constructibility real degrees is rigid, i.e. it does not have non-trivial automorphisms.

💡 Research Summary

The paper investigates the structure of constructibility real degrees in the side‑by‑side Sacks model, which is obtained by forcing over the constructible universe L with a countable‑support product of infinitely many Sacks forcings. The central object of study is the join‑semilattice (Dᶜ, ≤ᶜ) of constructibility degrees of reals, where a ≤ᶜ b iff a ∈ L(b). The authors establish three main structural results: (1) (Dᶜ, ≤ᶜ) is neither a meet‑semilattice, nor σ‑complete, nor a complete lattice; (2) it is rigid, i.e., it admits no non‑trivial automorphisms; (3) if a least or greatest real degree exists, then no other real degree can be defined within the structure.

The technical heart of the paper is a representation theorem (Theorem 5) that identifies each constructibility degree r with the set Ω(r) = {α < κ | s_α ≤ᶜ r}, where {s_α | α ∈ κ} are the generic Sacks reals added by the product forcing. The authors define a family R ⊆