Effective Reducibility for Statements of Arbitrary Quantifier Complexity with Ordinal Turing Machines

TL;DR

This paper extends effective reducibility concepts for set-theoretic statements of arbitrary complexity using ordinal Turing machines, comparing fundamental principles and characterizing HOD computationally.

Contribution

It generalizes effective reducibility to all quantifier complexities and introduces a computational perspective on set-theoretic principles and HOD.

Findings

Effective reducibility is strictly weaker than classical truth and OTM-realizability.

Characterizes HOD as sets computable relative to every effectivizer of -separation.

Introduces a variant of Weihrauch reducibility for set-theoretic statements.

Abstract

This paper is an extended version of our work in \cite{Ca2025}. We extend the concept of effective reducibility between statements of set theory with ordinal Turing machines (OTMs) explored in \cite{Ca2018} for $\Pi_{2}$-statements to statements of arbitrary quantifier complexity in prenex normal form and use this to compare various fundamental set-theoretical principles, including the power set axiom, the separation scheme, the collection scheme and the replacement scheme and various principles related to the notion of cardinality, with respect to effective reducibility. This notion of reducibility is both different from (i.e., strictly weaker than) classical truth and from the OTM-realizability of the corresponding implications. Along the way, we obtain a computational characterization of HOD as the class of sets that are OTM-computable relative to every effectivizer of $\Sigma_{2}$-separation. We also consider an associated variant or Weihrauch reducibility.