Broad Infinity and Generation Principles

This paper introduces Broad Infinity, a new and arguably intuitive axiom scheme. It states that broad numbers, which are three-dimensional trees whose growth is controlled, form a set. If the Axiom of Choice is assumed, then Broad Infinity is equivalent to the Ord-is-Mahlo scheme: every closed unbounded class of ordinals contains a regular ordinal. Whereas the axiom of Infinity leads to generation principles for sets and families and ordinals, Broad Infinity leads to more advanced versions of these principles. The paper relates these principles under various prior assumptions: the Axiom of Choice, the Law of Excluded Middle, and weaker assumptions.