Using results announced by Stevo Todorcevic we establish that if it is consistent that there is a supercompact cardinal then it is consistent that every locally compact perfectly normal space is paracompact. Modulo the large cardinal, this answers a question of S. Watson. We also solve a problem raised by the second author, proving that it is consistent with ZFC that every first countable hered...

In [6], we modified a coding device from [14] and the consistency proof of Martin’s Maximum from [3] to show that from a supercompact limit of supercompact cardinals one could force Martin’s Maximum to hold while the Pmax axiom (∗) fails. Here we modify that argument to prove a stronger fact, that Martin’s Maximum is consistent with the existence of a wellordering of the reals definable in H(א2...

We show that, assuming the consistency of a supercompact cardinal, the first (weakly) inaccessible cardinal can satisfy a strong form of a Löwenheim-Skolem-Tarski theorem for the equicardinality logic L(I), a logic introduced in [4] strictly between first order logic and second order logic. On the other hand we show that in the light of present day inner model technology, nothing short of a sup...

We show that from infinitely many supercompact cardinals one can force a model of ZFC where both the tree property and the stationary reflection hold at אω2+1.

We show that given ω many supercompact cardinals, there is a generic extension in which the tree property holds at אω2+1 and the SCH fails at אω2 .

Assuming the existence of a supercompact cardinal and an inaccessible above it, we construct a model of ZFC, in which all uncountable regular cardinals are inacces-