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

TL;DR

This work analyzes the join-semilattice of constructibility real degrees $(\mathcal{D}_c, \le_c)$ in the side-by-side Sacks model $\mathrm{L}[G]$, establishing a powerful representation theorem that identifies $(\mathcal{D}_c, \le_c)$ with a concrete system $(\mathcal{R}, \subseteq)$ built from the generic Sacks reals. The authors prove that the semilattice is rigid, not complemented, and not sigma-complete, and that, except for trivial least/greatest elements, no degree is definable without parameters. They further show that the set of Sacks degrees is amorphous and that no embedding of $(\mathcal{P}(\omega), \subseteq)$ into $(\mathcal{D}_c, \le_c)$ can be an ideal, revealing fundamental limits on representing the power set within this framework. The results hinge on a detailed fusion-based analysis and a key representation map $\Omega$, enabling transfer of combinatorial properties to the constructibility-degrees structure. Overall, the paper clarifies the rigid and highly non-absolute nature of real degrees in the infinite-sided Sacks setting and delineates sharp boundaries for definability and embeddability within this context.

Abstract

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.

Theorems & Definitions (47)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Lemma 6
• proof
• Proposition 7
• proof
• Corollary 8