Seminars & Colloquia Calendar

• Abstract:  Stronger tree properties capture the combinatorial essence of large cardinals. More precisely, for an inaccessible cardinal \(\kappa\), \(\kappa\) has the super, tree property (ITP) if and only if \(\kappa\) is supercompact. An old project in set theory is to get the tree property at every regular cardinal greater than \(\omega_1\). Even more ambitiously, can we get ITP at all regular cardinals above \(\omega_1\)? This would require many violations of the singular cardinal hypothesis (SCH), and leads to the question whether ITP implies SCH above. A positive answer would be an analogue of Solovay's theorem. We will show that consistently we can have ITP at some \(\lambda\) together with failure of SCH above \(\lambda\), for a non limit singular cardinal. The case of a limit singular cardinal is still open. We will also show that there is a model where ITP holds at the double successor of a singular and there are club many non internally unbounded models. This is another result in the direction of showing that ITP does not imply SCH above. Finally we discuss obtaining ITP at the successor of a singular cardinal.

This is joint work with Sherwood Hachtman.