Subtle and Ineffable Tree Properties

Abstract

In the style of the tree property, we give combinatorial principles that capture the concepts of the so-called subtle and ineffable cardinals in such a way that they are also applicable to small cardinals. Building upon these principles we then develop a further one that even achieves this for supercompactness. We show the consistency of these principles starting from the corresponding large cardinals. Furthermore we show the equiconsistency for subtle and ineffable. For supercompactness, utilizing the failure of square we prove that the best currently known lower bounds for consistency strength in general can be applied. The main result of the thesis is the theorem that the Proper Forcing Axiom implies the principle corresponding to supercompactness.