The Cichon Diagram for Degrees of Relative Constructibility

I just submitted my first paper: The Cichon Diagram for Degrees of Constructibility, on the ArXiv here, and I wanted to take the opportunity to write something about it here.

Abstract: Following a line of research initiated by Brendle, Brooke-Taylor, Ng and Nies, I describe a general framework for turning reduction concepts of relative computability into diagrams forming an analogy with the Cichon 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 Cichon 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.

In this paper I consider generalizations of the Cichon Diagram for reduction concepts. A reduction concept is a triple $(X, \sqsubseteq, 0)$ where $X$ is a non-empty set, $\sqsubseteq$ is a partial preorder on $X$ and $0 \in X$ is a distinguished element. An element $x \in X$ is called basic if $x \sqsubseteq 0$. Common examples include Turing reducibility with the basic elements being the computable sets, arithmetic reducibility with the basic elements being the $\emptyset$-definable subsets of $\mathbb N$ and degrees of constructibility with the basic elements being the constructible reals (in all these cases the underlying set is the reals). I show for a wide variety of reduction concepts on the reals, one can develop a generalization of the ideas underlying the study of the cardinal characteristics of the continuum by considering sets $\mathcal B_\sqsubseteq (R)$ and $\mathcal D_\sqsubseteq (R)$ for various relations $R$, where $\mathcal B_\sqsubseteq (R)$ is the set of elements which $\sqsubseteq$-build a witness to the fact that the basic reals are small with respect to $R$ and $\mathcal D_\sqsubseteq (R)$ is the set of elements which $\sqsubseteq$-build a witness to the fact that the basic reals are not big with respect to $R$. I show that for such reduction concepts one can always construct an analogue of the Cichon diagram. For example, below is the Cichon diagram for degrees of constructibility relative to a fixed inner model $W$. Recall that $x \leq_W y$ if and only if $x \in W[y]$. In the second part of the paper I focus on the case of $\leq_W$ and show that all of the arrows pictured above are consistently strict.

Theorem: The Cichon diagram for $\leq_W$ as shown above is complete in the sense that if $A$ and $B$ are two nodes in the diagram and there is not an arrow from $A$ to $B$ in the $\leq_W$-diagram then there is a forcing extension of $W$ where $A$ is not a subset of $B$.

The proof of this theorem involves studying how the diagram is affected by familiar forcing notions to add reals. Accompanying my arguments are versions of the diagram as affected by these forcing notions. For example, here are the versions for Sacks, Cohen and Hechler forcing.

In the end I show that there is one proper forcing over $W$ which simultaneously realizes all possible separations. This is the last main theorem proved in the paper. It’s formalized in GBC.

Theorem: Given any transitive inner model $W$ of ZFC, there is a proper forcing notion $\mathbb P$, such that in $W^\mathbb P$ all the nodes in the $\leq_W$-Cicho\’n diagram are distinct and every possibile separation is simultaneously realized. For more information check out the paper! Also, you can see a blog post about it on the page of my advisor, Joel David Hamkins, to whom I am incredibly grateful, here.