Discontinuous homomorphisms on C(X) with the negation of CH and a weak forcing axiom

Abstract In this paper, I introduce the properties and for forcing notions and show that it is consistent that the forcing axiom for forcing notions holds, the continuum hypothesis fails, and an ultrapower of the field of reals has the property . This provides a partial solution to H. Woodin's question concerning the existence of discontinuous homomorphisms on the Banach algebra of all complex‐valued continuous functions on a compact space. Furthermore, we prove that the uniformization of a coloring of a ladder system on a stationary–costationary set is an example of an forcing notion. As a corollary, it is consistent that a nonfree Whitehead group exists, the continuum hypothesis fails, and an ultrapower of the field of reals has the property .