The Cichoń Diagram for Degrees of Relative Constructibility

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 Cichoń 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 Cichoń 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.

arXiv comment25 pages, 14 figures. Fourth version adds some citations, cutting down on standard proofs and includes some additional observations based on very helpful comments from an anonymous referee. Now accepted at Mathematical Logic Quarterly