UNREACHABILITY OF INDUCTIVE-LIKE POINTCLASSES IN L(R)

In [3], Hjorth proved from $ZF + AD + DC$ that there is no sequence of distinct Σ ¹₂ sets of length δ ¹₂ . Sargsyan [11] extends Hjorth’s technique to show there is no sequence of distinct Σ ¹₂ₙ sets of length δ ¹₂ₙ . Sargsyan conjectured an analogous property is true for any regular Suslin pointclass in L(R) —i.e., if κ is a regular Suslin cardinal in L(R) , then there is no sequence of distinct κ -Suslin sets of length κ ⁺ in L(R) . We prove this in the case that the pointclass S(κ ) is inductive-like.