Epsilon0hoz

epsilon0hoz is a hypothetical ordinal-like construct that appears in certain mathematical folklore and speculative discussions of ordinal analysis. It is intended as a higher-stage extension of Cantor's epsilon number ε0, used to probe deeper transfinite closure properties.

Definition: In informal treatments, epsilon0hoz is described as the least ordinal greater than ε0 that is closed under a hoz-operator H. The hoz-operator is a abstract closure mechanism defined by iterating a fixed family of operations, including ω-exponentiation and a nondecreasing combinator. By applying H through transfinite stages starting from ε0, one arrives at the limit commonly identified as epsilon0hoz. The exact formalism of H can vary between presentations.

Properties: Epsilon0hoz is characterized as a countable ordinal in these informal accounts. It is viewed as a fixed point of the hoz-operator in the sense of the stated closure, and it is strictly larger than ε0. Because the construction depends on the chosen family of operations that define H, different accounts may yield variant but related notions of epsilon0hoz.

Context and usage: The notion is primarily used in expository discussions to illustrate how transfinite closures can generate new fixed points in ordinal notation. It is encountered in fictional or pedagogical contexts to motivate questions about the limits of iterative constructions, and it does not have a single universally accepted formalization.

See also: ε0, fixed point, ordinal notation, fast-growing hierarchy.