Let $\phi$ be an acceptable programming system. For every recursive function $f$, let $(f)={x:\phi_x=\phi_{f(x)}}$ (f)=\{x:\phi_x=\phi_{f(x)}\}$ the set of fixed points of $f$. Now, suppose that $f$ and $g$ are recursive functions such that $(f)$ and $(g)$ are disjoint sets. The question is: are there recursive functions $F$ and $G$ such that $(F)\subseteq(f)$, $(G)\subseteq(g)$ and $\phi_{F(x)}\neq\phi_{G(x)}$, for all $x$?
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Disjoint sets of fixed pointsLet $\phi$ be an acceptable programming system. For every recursive function $f$, let $(f)={x:\phi_x=\phi_{f(x)}}$ the set of fixed points of $f$. Now, suppose that $f$ and $g$ are recursive functions such that $(f)$ and $(g)$ are disjoint sets. The question is: are there recursive functions $F$ and $G$ such that $(F)\subseteq(f)$, $(G)\subseteq(g)$ and $\phi_{F(x)}\neq\phi_{G(x)}$, for all $x$?
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