Predicative fragment of the transfinite induction scheme (W2)

Determine whether there exists a formally specified, predicatively justified class of third‑order formulas for which the unrestricted transfinite induction scheme (W2) over the global well‑ordering relation \(\prec\) can be asserted in Weaver’s system CM without impredicativity, thereby yielding a restricted extension of CM that is philosophically acceptable within mathematical conceptualism.

Background

Weaver proposed augmenting CM with axioms for a global well‑ordering on second‑order objects together with a transfinite induction scheme. For conceptualist reasons, he intended the induction scheme not to apply to arbitrary third‑order formulas, but he did not provide a formal description of an acceptable restricted class.

The paper highlights that the lack of a precise, predicatively acceptable fragment for transfinite induction remains unresolved. Clarifying such a fragment would bridge philosophical constraints with technical utility, enabling a principled extension of CM that avoids impredicativity while still supporting transfinite arguments.