Assume that M is a transitive model 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 $2^{\aleph _0}>\aleph _2$ 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.

We put in print a classical result that states that for most purposes, there is no harm in assuming the existence of saturated models in model theory. The presentation is aimed for model theorists with only basic knowledge of axiomatic set theory.

Let κ be the cardinality of some saturated model of Peano Arithmetic. There is a set of ${2^{{\aleph _0}}}$ saturated models of PA, each having cardinality κ, such that whenever M and N are two distinct models from this set, then Aut(${\cal M}$) ≇ Aut ($${\cal N}$$).