Philip D. Welch

We show that the predicate “ x is the power set of y ” is ‐definable, if V = L[E] is an extender model constructed from a coherent sequences of extenders, provided that there is no inner model with a Woodin cardinal. Here is a predicate true of just the infinite cardinals. From this we conclude: the validities of second order logic are reducible to , the set of validities of the Härtig quantifier logic. Further we show that if no L[E] model has a cardinal strong up to one of its ℵ‐fixed points, and , the Löwenheim number of this logic, is less than the least weakly inaccessible δ, then (i) is a limit of measurable cardinals of K, and (ii) the Weak Covering Lemma holds at δ.