arXiv subjects
Publications and source records attributed to Kaethe Minden.
It is shown that the resurrection axiom and the maximality principle may be consistently combined for various iterable forcing classes. The extent to which resurrection and maximality overlap is explored via the local maximality principle.
Let $D$ be an infinite discrete set of measurable cardinals. It is shown that generalized Prikry forcing to add a countable sequence to each cardinal in $D$ is subcomplete. To do this it is shown that a simplified version of generalized Prikry forcing which adds a point below each cardinal in $D$, called generalized diagonal Prikry forcing, is subcomplete. Moreover, the generalized diagonal Prikry forcing associated to $D$ is subcomplete above $μ$, where $μ$ is any regular cardinal below the first limit point of $D$.
Regarding neighbor balance, we consider natural generalizations of $D$-complete Latin squares and Vatican squares from the finite to the infinite. We show that if $G$ is an infinite abelian group with $|G|$-many square elements, then it is possible to permute the rows and columns of the Cayley table to create an infinite Vatican square. We also construct a Vatican square of any given infinite order that is not obtainable by permuting the rows and columns of a Cayley table. Regarding orthogonality, we show that every infinite group $G$ has a set of $|G|$ mutually orthogonal orthomorphisms and hence there is a set of $|G|$ mutually orthogonal Latin squares based on $G$. We show that an infinite group $G$ with $|G|$-many square elements has a strong complete mapping; and, with some possible exceptions, infinite abelian groups have a strong complete mapping.
We investigate properties of trees of height $ω_1$ and their preservation under subcomplete forcing. We show that subcomplete forcing cannot add a new branch to an $ω_1$-tree. We introduce fragments of subcompleteness which are preserved by subcomplete forcing, and use these in order to show that certain strong forms of rigidity of Suslin trees are preserved by subcomplete forcings. Finally, we explore under what circumstances subcomplete forcing preserves Aronszajn trees of height and width $ω_1$. We show that this is the case if CH fails, and if CH holds, then this is the case iff the bounded subcomplete forcing axiom holds. Finally, we explore the relationships between bounded forcing axioms, preservation of Aronszajn trees of height and width $ω_1$ and generic absoluteness of $Σ^1_1$-statements over first order structures of size $ω_1$, also for other canonical classes of forcing.
I survey an array of topics in set theory in the context of a novel class of forcing notions: subcomplete forcing. Subcompleteness was originally defined by Ronald Jensen. I have attempted to make the subject somewhat more approachable to set theorists, while showing various properties of subcomplete forcing which one might desire of a forcing class, drawing comparisons between subcomplete forcing and countably closed forcing. In particular, I look at the interaction between subcomplete forcing and $ω_1$-trees, preservation properties of subcomplete forcing, the subcomplete maximality principle, the subcomplete resurrection axiom, and show that generalized diagonal Prikry forcing is subcomplete.
We introduce the split principles and show that they bear tight connections to large cardinal properties such as inaccessibility, weak compactness, subtlety, almost ineffability and ineffability, as well as classical combinatorial objects such as Aronszajn trees, Souslin trees or square principles. We exhibit correspondences between certain split principles and splitting numbers at uncountable cardinals.