Wand/set theories: A realization of Conway's mathematicians' liberation movement, with an application to Church's set theory with a universal set

TL;DR

This paper introduces a novel set-theoretic framework inspired by a playful 'wand' analogy, demonstrating its equivalence to ZF-like theories and applying it to Church's set theory with a universal set, thus bridging foundational ideas.

Contribution

It formalizes a new class of set theories based on wand-augmented iterative processes, linking Conway's liberation movement with Church's set theory, and establishing their equivalence to classical ZF-like theories.

Findings

Any loosely constructive wand-based set theory is equivalent to a ZF-like theory.

The framework realizes Conway's Mathematicians' Liberation Movement.

Connections are established with Church's set theory with a universal set.

Abstract

Consider a variant of the usual story about the iterative conception of sets. As usual, at every stage, you find all the (bland) sets of objects which you found earlier. But you also find the result of tapping any earlier-found object with any magic wand (from a given stock of magic wands). By varying the number and behaviour of the wands, we can flesh out this idea in many different ways. This paper's main Theorem is that any loosely constructive way of fleshing out this idea is synonymous with a ZF-like theory. This Theorem has rich applications; it realizes John Conway's (1976) Mathematicians' Liberation Movement; and it connects with a lovely idea due to Alonzo Church (1974).