Let $H$ be an infinite dimensional separable complex Hilbert space. Denote by $\mathcal{B}(H)$ the C*-algebra of bounded operators on $H$, $\mathcal{K}(H)$ the ideal of compact operators on $H$, and $\mathcal{C}(H)=\mathcal{B}(H)/\mathcal{K}(H)$ the Calkin algebra. The collection of projections $\mathcal{P}(\mathcal{C}(H))$, i.e., self-adjoint idempotents, in $\mathcal{C}(H)$ form a partially ordered set in the usual way, namely $p\leq q$ if and only if $pq=p$.

It is my understanding that $(\mathcal{P}(\mathcal{C}(H)),\leq)$ does not form a lattice, i.e., meets and joins need not exist in general. (This would mean that the Calkin algebra is another example for C*-algebras with bizzarre structure of projections.)

This fact is mentioned in Hadwin's paper "Maximal Nests in the Calkin Algebra" (PAMS, 1998), and there is a sketch of a proof (credited to Weaver) in Farah's notes "Set theory and operator algebras" from the Appalachian Set Theory Workshop, however I've never quite understood this proof.

Would someone be able to explain/sketch this result?