Black Boxes

TL;DR

This paper explores the existence and properties of 'Black Boxes' in set theory, demonstrating their construction via game-theoretic methods and linking them to ideal non-saturation and diamond principles.

Contribution

It provides a comprehensive analysis of Black Boxes, including their construction, properties, and connections to other set-theoretic concepts, extending previous work with new systematic approaches.

Findings

Black Boxes can be constructed using game-theoretic methods.

Existence of Black Boxes relates to non-saturation of natural ideals.

Connections established between Black Boxes and diamond principles.

Abstract

We shall deal comprehensively with Black Boxes, the intention being that provably in ZFC we have a sequence of guesses of extra structure on small subsets, where the guesses are pairwise almost disjoint; by this we mean they have quite little interaction, and are far apart but together are dense. We first deal with the simplest case, where the existence comes from winning a game by just writing down the opponent's moves. We show how it helps when instead of orders we have trees with boundedly many levels, having freedom in the last. After this we quite systematically look at existence of black boxes, and make connection to non-saturation of natural ideals and diamonds on them.