Discontinuous Homomorphisms of $C(X)$ with $2^{\aleph_0}>\aleph_2$

Abstract

Assume that $M$ is a c.t.m. of $ZFC+CH$ containing a simplified $(\omega_1,2)$-morass, $P\in M$ is the poset adding $\aleph_3$ generic reals and $G$ is $P$-generic over $M$. In $M$ we construct a function between sets of terms in the forcing language, that interpreted in $M[G]$ is an $\mathbb R$-linear order-preserving monomorphism from the finite elements of an ultrapower of the reals, over a non-principal ultrafilter on $\omega$, into the Esterle algebra of formal power series. Therefore it is consistent that $2^{\aleph_0}=\aleph_3$ and, for any infinite compact Hausdorff space $X$, there exists a discontinuous homomorphism of $C(X)$, the algebra of continuous real-valued functions on $X$. For $n\in \mathbb N$, If $M$ contains a simplified $(\omega_1,n)$-morass, then in the Cohen extension of $M$ adding $\aleph_n$ generic reals there exists a discontinuous homomorphism of $C(X)$, for any infinite compact Hausdorff space $X$.