The Cicho\'n Diagram for Degrees of Relative Constructibility

TL;DR

This paper develops a framework to create diagrams analogous to the Cicho'n diagram for degrees of relative constructibility, revealing new insights and differences in the structure of degrees relative to inner models.

Contribution

It introduces a general method for constructing Cicho'n-like diagrams from reduction notions and applies it to degrees of constructibility relative to an inner model, highlighting both similarities and differences with classical theory.

Findings

Constructed a robust diagram for degrees of relative constructibility.

Identified analogies and differences with the classical Cicho'n diagram.

Proposed a new axiom about the complexity of the constructibility diagram.

Abstract

Following a line of research initiated in \cite{BBNN}, I describe a general framework for turning reduction concepts of relative computability into diagrams forming an analogy with the Cicho\'n diagram for cardinal characteristics of the continuum. I show that working from relatively modest assumptions about a notion of reduction, one can construct a robust version of such a diagram. As an application, I define and investigate the Cicho\'n Diagram for degrees of constructibility relative to a fixed inner model $W$. Many analogies hold with the classical theory as well as some surprising differences. Along the way I introduce a new axiom stating, roughly, that the constructibility diagram is as complex as possible.