Disclaimer: I posted a questions on constructor theory here a few days ago but received two closing votes, I guess because it consisted of several subquestions, so I deleted it and now try to focus on a single issue.

I'm reading Constructor Theory Of Probability and became a bit confused about the notion of coarsening:

This seems very natural to me, $X'$ is a coarsening of $X$ iff there exists a partition $C$ on $X$ such that $X'=\{\cup c:c\in C\}$. The second part I understand in the sense that arbitrary maps $f:S\to X$ on the state space are associated with the variables $X_f:=\{f^{-1}(x):x\in X\}$, and if $f_1$, $f_2$ map into any ring $R$, we have $X_{f_1+f_2}$ and $X_{f_1f_2}$ being coarsenings of $X_{(f_1,f_2)}$ (where $(f_1,f_2)$ is interpreted as map into $R^2$).

But in the following part, they seem to use a different notion:

If $\forall x\in X:x\cap y=\emptyset$ holds, as indicated here, which is equivalent to $x\cap(\cup X)=\emptyset$, how can some coarsening of $X$ be sharp in $y$, as $\cup X=\cup X'$ for any coarsening $X'$? (The definition of a $X$ being sharp in $a$ is $\exists x\in X:a\subseteq x$.)

As far as their example with the photon number goes, they earlier claim 'The $z$-component of the spin is a variable, represented as the set of two intrinsic attributes: that of the $z$-component of the spin being $1/2$ and $−1/2$', so here I understand $X$ to equal $\{\{|n\rangle\}:n\in\omega\}$ (the eigenspaces of the projector). Then it is true that $(1/\sqrt{2})(|0\rangle+|1\rangle)$ lies in the eigenspace (what do they mean by $+1$-eigenspace?) of $|0\rangle\langle 0|+|1\rangle\langle1|$ but if we interpret projectors as variables this way, we obtain a different meaning of addition, especially $X_{P_1+P_2}$ does not equal $X_{P_1}+X_{P_2}$ in the way I understand it!

Would it be appropriate to define a coarsening $X'$ of $X$ by $\forall x\in X:\exists x'\in X':x\subseteq x'$, or equivalently, whenever $X$ is sharp in some attribute $a$, so must be $X'$? Because that's mostly compatible with those two paragraphs.


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