Subtle and Ineffable Tree Properties

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.