We construct an explicit transfinite hierarchy of reflective theories indexed by a binary tree with controlled branching. We show that the set of infinite paths through this tree is Π₁¹-complete, while its restrictions by maximum branching degree form a strict hierarchy from Π₁⁰-complete to Π₁¹-complete sets. The construction is based on an operator that simultaneously adds both a consistency statement and a full reflection scheme. As a corollary, we obtain that the set of sentences independent of all theories in the hierarchy (called the Higher Set of Undecidable Types, HSUT) is also Π₁¹-complete, formalizing the notion of "absolute silence" in the foundations of mathematics.
Daniel Osipenkov (Thu,) studied this question.
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