σ-Separation in Descriptive Set Theory

• σ-separation is the assertion that any two disjoint sets in a definable pointclass can be separated by a Δ-set, clarifying definability within the projective hierarchy.
• Forcing methods leveraging almost-disjoint coding and Suslin tree sequences are used to achieve Σ¹₃-separation while ensuring the failure of Π¹₃-reduction.
• These results expose a separation-reduction dichotomy in inner models like L and Mₙ, prompting further inquiry into higher-level regularity phenomena.

σ-separation refers to the separation property for definable pointclasses within the projective hierarchy, specifically in the context of descriptive set theory and inner model theory. Given a pointclass Γ, typically one of the projective classes Σ¹ₙ or Π¹ₙ, σ-separation (occasionally written as Σ-separation or Σ¹ₙ-separation for a fixed n) is the assertion that any two disjoint sets in Γ can be separated by a Δ-set at the same level. This property is central to understanding regularity phenomena, definability, and the effect of various set-theoretic constructions on the projective hierarchy, particularly in the presence of large cardinals and forcing.

1. Projective Hierarchy and Pointclass Separation

The projective hierarchy organizes sets of reals by the complexity of their definability, beginning with the analytic (Σ¹₁) and coanalytic (Π¹₁) sets, then iteratively applying projection and complementation:

• Σ¹₁ sets: Continuous images of Borel sets (analytic sets).
• Π¹₁ sets: Complements of Σ¹₁ sets.
• Inductively, for $n \geq 1$:
  • $\Sigma^1_{n+1}$ consists of projections of $\Pi^1_n$ sets.
  • $\Pi^1_{n+1}$ consists of complements of $\Sigma^1_{n+1}$ sets.

A pointclass Γ is said to have the separation property (σ-separation) if, for any two disjoint sets $A, B \in \Gamma$, there is $C \subseteq \mathbb{R}$ with $C \in \Gamma \cap \bar{\Gamma}$ (i.e., $C$ is at the corresponding Δ-level) such that $A \subseteq C$ and $B \subseteq \mathbb{R} \setminus C$. Specifically, Σ¹ₙ₊₃-separation asserts:

$$\forall A, B \subseteq \mathbb{R}\; \Bigl( A, B \in \Sigma^1_{n+3}\; \wedge\; A \cap B = \varnothing\; \Longrightarrow\; \exists C \in \Delta^1_{n+3}\; [A \subseteq C,\; B \subseteq \mathbb{R} \setminus C] \Bigr)$$

2. Forcing Methods and the Construction of σ-Separation

In $L$ (the constructible universe), it is possible to force Σ¹₃-separation to hold while simultaneously making Π¹₃-reduction fail. The methodology employs a combination of definable independent families of Suslin trees and interspersed coding via almost-disjoint sets and Cohen forcing. The construction proceeds by:

• Enumerating all pairs of disjoint Σ³-sets by parameters in $L$.
• At each stage or according to a bookkeeping function, forcing either with trivial forcing or specialized coding iterations such as $$\mathrm{Code}(x, m, k, i) = (\mathrm{Add}(\omega, 1))*\mathrm{AD}(Y)$$, where $\mathrm{AD}(Y)$ is an almost-disjoint coding forcing.
• Utilizing mixed-support iterations ($P_{\omega_1}$), ensuring the preservation of $\omega_1$ and the Continuum Hypothesis, and maintaining the ccc/σ-closed properties needed for control over the generic extension.

By this process, every disjoint pair of $\Sigma^1_3$ sets are separated by a $\Delta^1_3$ set definable via a real parameter introduced in the extension, ensuring the full effectivity of Σ¹₃-separation (Hoffelner, 2023).

3. Failure of Π-Reduction and the Separation-Reduction Dichotomy

Reduction is a dual concept to separation. A pointclass Γ has the reduction property if any pair of sets $B_0, B_1 \in \Gamma$ can be partitioned into disjoint subsets $R_0 \subseteq B_0$, $R_1 \subseteq B_1$, both in Γ, such that $R_0 \cup R_1 = B_0 \cup B_1$.

The construction outlined above also enables a diagonalization at even stages:

• For candidate Π³-set pairs, the forcing ensures that certain reals are coded so that splitting (i.e., reduction) by Π³-sets becomes impossible.
• The diagonalization is robust under further allowable extensions and ensures a provable failure of Π¹₃-reduction. Thus, in the final model, there exist lightface Π¹₃-sets $B_0, B_1$ for which no pair $(R_0, R_1)$ of Π¹₃-sets can satisfy the simultaneous containment and covering conditions of the reduction property.

Significantly, this establishes that Σ¹₃-separation can be forced to hold in a model where Π¹₃-reduction provably fails, thereby dissecting the previously presumed close alignment between separation and reduction properties at successive levels of the hierarchy (Hoffelner, 2023).

4. Generalization to Inner Models with Woodin Cardinals

The method is extendable to canonical inner models $M_n$ containing $n$ Woodin cardinals. By adapting the machinery:

• The construction uses the $M_n$-version of a definable independent Suslin-tree sequence and $M_n$-internal almost-disjoint coding.
• Forcing iterations of length $\omega_1^{M_n}$ operate as in $L$, preserving the relevant cardinals and maintaining absoluteness for the formulas in question.

In the resulting $M_n[G]$:

• $\Sigma^1_{n+3}$-separation holds.
• $\Pi^1_{n+3}$-reduction fails.

Absoluteness lemmas guarantee that resourceful definability and regularity properties are preserved and that no unwanted codes or realizations are inadvertently introduced by forcing (Hoffelner, 2023).

5. Implications and Open Problems

The separation of Σ¹ₙ₊₃-separation from Π¹ₙ₊₃-reduction achieved in these models is the first of its kind, challenging prevailing assumptions about the coincidence of these properties in $L$ and under the Axiom of Projective Determinacy (PD). This demonstrates that, even in the context of large cardinals, the descriptive set-theoretic landscape at high levels of definitional complexity can manifest divergent regularity phenomena.

Prominent open questions emerging from this development include:

• Whether one can force $\Sigma^1_4$-separation while $\Pi^1_4$-reduction fails over $L$.
• For $n \geq 3$, whether there exists a model in which $\Sigma^1_n$-separation holds but $\Pi^1_n$-reduction fails.
• The possibility of simultaneously forcing both $\Sigma^1_3$-separation and $\Pi^1_3$-separation.

These problems remain central to understanding the precise interaction and possible independence of separation and reduction at various points in the projective hierarchy (Hoffelner, 2023).

6. Summary Table: Separation and Reduction in the Projective Hierarchy

Level (n)	Σ¹ₙ₊₃-Separation	Π¹ₙ₊₃-Reduction
$n+3$ in $L$	Can be forced to hold	Can be forced to fail
$n+3$ in $M_n$	Can be forced to hold	Can be forced to fail

The table summarizes that, both in $L$ and in the inner models $M_n$ with $n$ Woodin cardinals, there exist forcing extensions witnessing the separation of these two principles at level $n+3$, a fundamental advancement in higher descriptive set theory (Hoffelner, 2023).