Broad Infinity and Generation Principles

TL;DR

The paper introduces Broad Infinity, a new set-theoretic axiom scheme that generalizes the concept of set formation, connecting it to Mahlo's principle and enabling the construction of Grothendieck universes without relying on ordinals.

Contribution

It proposes Broad Infinity as a novel axiom scheme, establishing its equivalence to Mahlo's principle and demonstrating its power to generate universes and sets under various choice assumptions.

Findings

Broad Infinity is equivalent to Mahlo's principle under AC or WISC.

It enables the construction of Grothendieck universes without ordinal detours.

In the absence of choice, it implies the sethood of derivations from rubrics.

Abstract

We introduce Broad Infinity, a new set-theoretic axiom scheme based on the slogan "Every time we construct a new element, we gain a new arity." It says that three-dimensional trees whose growth is controlled by a specified class function form a set. Such trees are called "broad numbers". Assuming AC (the axiom of choice), or at least the weak version known as WISC (Weakly Initial Set of Covers), we show that Broad Infinity is equivalent to Mahlo's principle, which says that the class of all regular limit ordinals is stationary. Assuming AC or WISC, Broad Infinity also yields a convenient principle for generating a subset of a class using a "rubric" (family of rules). This directly gives the existence of Grothendieck universes, without requiring a detour via ordinals. In the absence of choice, Broad Infinity implies that the derivations of elements from a rubric form a set. This yields the existence of Tarski-style universes. Additionally, we reveal a pattern of resemblance between "Wide" principles, that are provable in ZFC, and "Broad" principles, that go beyond ZFC. Note: this paper uses a base theory that is weaker than ZF but includes classical first-order logic and Replacement.