Ordered Black Boxes: Existence

TL;DR

This paper introduces ordered black boxes related to pcf theory, focusing on predicting bounds in partial orders and exploring their existence, including trivial cases and propagation properties.

Contribution

It establishes the existence of ordered black boxes linked to pcf theory and examines their properties, including trivial cases and downward propagation.

Findings

Existence of ordered black boxes connected to pcf theory.

Trivial cases of ordered black boxes are identified.

Propagation of ordered black boxes downward is demonstrated.

Abstract

We defined ordered black boxes in which for a partial $J$ we try to predict just a bound in $J$ to a function restricted to $C_\alpha$. The existence results are closely related to pcf, propagating downward. We can start with trivial cases.