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Broad Infinity and Generation Principles

Paul Blain Levy

TL;DR

The paper introduces Broad Infinity, a new axiom scheme that extends a base theory weaker than ZF by guaranteeing the existence of sets formed by progressively introducing new arities as elements are constructed. It establishes a tight link between Simple Broad Infinity and Mahlo's principle under AC or WISC, and shows that Broad Set Generation yields Grothendieck universes while Broad Derivation Set yields Tarski-style universes, thereby providing practical universe-construction tools without detouring through ordinals. The work also analyzes the corresponding Wide Principles and demonstrates a systematic pattern of resemblance between Wide (ZF-provable) and Broad (beyond ZFC) principles, including the role of rubrics and derivations in generating subsets and universes. In the absence of full Choice, Broad Infinity implies a derivation-set principle that still yields universe-like constructions, highlighting robustness under weaker choice principles. The framework unifies set-theoretic and category-theoretic methods, offering a coherent path from inductive chains and fixpoints to the existence of large structures, and suggests promising directions for further exploration of Broad ZF and related systems.

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.

Broad Infinity and Generation Principles

TL;DR

The paper introduces Broad Infinity, a new axiom scheme that extends a base theory weaker than ZF by guaranteeing the existence of sets formed by progressively introducing new arities as elements are constructed. It establishes a tight link between Simple Broad Infinity and Mahlo's principle under AC or WISC, and shows that Broad Set Generation yields Grothendieck universes while Broad Derivation Set yields Tarski-style universes, thereby providing practical universe-construction tools without detouring through ordinals. The work also analyzes the corresponding Wide Principles and demonstrates a systematic pattern of resemblance between Wide (ZF-provable) and Broad (beyond ZFC) principles, including the role of rubrics and derivations in generating subsets and universes. In the absence of full Choice, Broad Infinity implies a derivation-set principle that still yields universe-like constructions, highlighting robustness under weaker choice principles. The framework unifies set-theoretic and category-theoretic methods, offering a coherent path from inductive chains and fixpoints to the existence of large structures, and suggests promising directions for further exploration of Broad ZF and related systems.

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.

Paper Structure

This paper contains 76 sections, 84 theorems, 21 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Broad Infinity vs Mahlo's Principle
  3. Broad Infinity in a Nutshell
  4. Regular Limits and Stationary Classes
  5. Two Principles from the Literature
  6. Limitations of Mahlo's Principle
  7. Motivating Broad Infinity
  8. Goals and Structure of the Paper
  9. Plausible vs Useful
  10. Urelements and Non-Well-Founded Membership
  11. Weak Choice Principle
  12. Wide vs Broad
  13. Summary of Goals
  14. Related Work
  15. Structure of Paper
  16. ...and 61 more sections

Key Result

Proposition 3.1

Let $C$ be a class. Any sets $A,B$ and surjection $f \! : \! A \rightarrow B$ yield an injective function $C^f \! : \! C^B \rightarrow C^A$ sending $[x_b]_{b \in B}$ to $[x_{f(a)}]_{a \in A}$. Its range is the class of all $A$-tuples $y$ such that, for all $a,a' \!\in\! A$ with the same $f$-image, w

Figures (2)

  • Figure 1: Diagram of theories, each extending the base theory
  • Figure 2: Visualization of a simple wide number

Theorems & Definitions (216)

  • Definition 1.1
  • Definition 1.2
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Definition 3.4
  • Proposition 3.5
  • ...and 206 more