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

Abstract

In this paper, I introduce the properties ${\mathrm{EPC}}_{{\aleph }_1}$ and $\mathrm{ProjCes}\left(E\right)$ for forcing notions and show that it is consistent that the forcing axiom for ${\mathrm{EPC}}_{{\aleph }_1}+\mathrm{ProjCes}\left(E\right)$ forcing notions holds, the continuum hypothesis fails, and an ultrapower of the field of reals has the property ${\beta }_1$. 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 $E$ is an example of an ${\mathrm{EPC}}_{{\aleph }_1}+\mathrm{ProjCes}({\omega }_1\setminus E)$ 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 ${\beta }_1$.