The Hanf number of second order logic is the least cardinal such that every sentence of second order logic that has a model of power at least has arbitrarily large models. Let denote the power-set operation and T the Kripke-Platek axioms in the language , augmented with the power-set axiom and the axiom that every well-ordering is isomorphic to an ordinal. Consider the following cardinals:
(a) The Hanf-number of second order logic,
(b) The least cardinal such that whenever with , then
(c) The least cardinal such that whenever and , then ; and whenever , , and , then
It is proved in T that if any of the cardinals (a)-(c) exists, they all exist and are equal. The result complements and builds on related results by J. Barwise [J. Symb. Logic 37, 588-594 (1972; Zbl 0281.02020)] and H. Friedman [ibid. 39, 318-324 (1974; Zbl 0293.02039)].