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DISCONTINUOUS HOMOMORPHISMS OF
$C(X)$ WITH 
$2^{\aleph _0}>\aleph _2$
- Part of
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- Journal:
- The Journal of Symbolic Logic / Volume 89 / Issue 2 / June 2024
- Published online by Cambridge University Press:
- 15 April 2024, pp. 665-696
- Print publication:
- June 2024
-
- Article
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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.
