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A long standing project in set theory is to analyze how much compactness can be obtained in the universe. Compactness is the phenomenon where if a certain property holds for all small substructures of an object, then it holds for the entire object. Compactness properties of particular interest are combinatorial principles that follow from large cardinals, but can be forced to hold at successors. Key examples include (in order of increasing strength) failure of squares, the tree property, and the ineffable tree property (ITP). These principles “capture” the combinatorial essence of certain large cardinals. At an inaccessible cardinal, the tree property is equivalent to weak compactness; ITP is equivalent to supercompactness. Forcing these principles at successors tells us to what extent small cardinals can behave like large cardinals. An old question of Magidor addressing these issues is: can we get principles like the tree property or ITP simultaneously for every regular cardinal greater than ω1? A positive answer would require many failures of SCH. In this paper we focus on ITP, the strongest of our key examples, and its relation to singular cardinal combinatorics. This is of particular interest because failure of SCH is an example of anticompactness, and so it is difficult to combine it with principles like ITP.

The ineffable tree property and failure of the singular cardinals hypothesis