WSPC Proceedings -9in x 6inBrendleBrookeNgNies˙Cichon˙recursion˙theory page 2 2 We present an analogy between cardinal characteristics from set theory and highness properties from computability theory, which specify a sense in which a Turing oracle is computationally strong. While this analogy was first studied explicitly by Rupprecht (Effective correspondents to cardinal characteristics in Cichoń's diagram, PhD thesis, University of Michigan, 2010), many prior results can be viewed from this perspective. After a comprehensive survey of the analogy for characteristics from Cichoń's diagram, we extend it to Kurtz randomness and the analogue of the Specker-Eda number.
We develop a version of Cichoń's diagram for cardinal invariants on the generalized Cantor space 2 κ or the generalized Baire space κ κ where κ is an uncountable regular cardinal. For strongly inaccessible κ, many of the ZFC-results about the order relationship of the cardinal invariants which hold for ω generalize; for example we obtain a natural generalization of the Bartoszyński-Raisonnier-Stern Theorem. We also prove a number of independence results, both with < κ-support iterations and κ-support iterations and products, showing that we consistently have strict inequality between some of the cardinal invariants.
We give a sharper version of a theorem of Rosický, Trnková and Adámek [13], and a new proof of a theorem of Rosický [12], both about colimits in categories of structures. Unlike the original proofs, which use category-theoretic methods, we use set-theoretic arguments involving elementary embeddings given by large cardinals such as α-strongly compact and C(n)-extendible cardinals.
We use a reverse Easton forcing iteration to obtain a universe with a definable wellordering, while preserving the GCH and proper classes of a variety of very large cardinals. This is achieved by coding using the principle ♦ * κ + at a proper class of cardinals κ. By choosing the cardinals at which coding occurs sufficiently sparsely, we are able to lift the embeddings witnessing the large cardinal properties without having to meet any non-trivial master conditions.
We provide a model where u(κ) = κ + < 2 κ for a supercompact cardinal κ.[10] provides a sketch of how to obtain such a model by modifying the construction in [6]. We provide here a complete proof using a different modification of [6] and further study the values of other natural generalizations of classical cardinal characteristics in our model. For this purpose we generalize some standard facts that hold in the countable case as well as some classical forcing notions and their properties.
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