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As is well-known, ZFC proves the equipotency of $\mathbb{R}$ and $\mathcal{P}(\mathbb{Q}).$ Is there a nice characterization of those linearly ordered sets $L$ which, like $\mathbb{Q}$, have the property that their completion by Dedekind-cuts $\mathcal{C}(L)$ is equipotent with their powerset $\mathcal{P}(L)$?

Edit. At the risk of introducing a non-standard concept into a perfectly standard question, I am especially interested in the case where $L$ is unbounded and "locally homogeneous," in the sense that for all $x,y \in L$ and all $x',y' \in L,$ we have that if $x<y$ and $x'<y'$, then the interval $(x,y)$ is isomorphic to the interval $(x',y')$. (The purpose of this condition is just to cut out all the crazy mish-mash orders involving mixtures of scattered and dense linear orders with complete and incomplete segments all smashed together into some crazy pattern. See Arthur's answer here for an example of a homogeneous order different from $\mathbb{R}$ and $\mathbb{Q}$).

As is well-known, ZFC proves the equipotency of $\mathbb{R}$ and $\mathcal{P}(\mathbb{Q}).$ Is there a nice characterization of those linearly ordered sets $L$ which, like $\mathbb{Q}$, have the property that their completion by Dedekind-cuts $\mathcal{C}(L)$ is equipotent with their powerset $\mathcal{P}(L)$?

Edit. At the risk of introducing a non-standard concept into a perfectly standard question, I am especially interested in the case where $L$ is unbounded and "locally homogeneous," in the sense that for all $x,y \in L$ and all $x',y' \in L,$ we have that if $x<y$ and $x'<y'$, then the interval $(x,y)$ is isomorphic to the interval $(x',y')$. (The purpose of this condition is just to cut out all the crazy mish-mash orders involving mixtures of scattered and dense linear orders with complete and incomplete segments all smashed together into some crazy pattern. See Arthur's answer here for an example of a homogeneous order different from $\mathbb{R}$ and $\mathbb{Q}$).

As is well-known, ZFC proves the equipotency of $\mathbb{R}$ and $\mathcal{P}(\mathbb{Q}).$ Is there a nice characterization of those linearly ordered sets $L$ which, like $\mathbb{Q}$, have the property that their completion by Dedekind-cuts $\mathcal{C}(L)$ is equipotent with their powerset $\mathcal{P}(L)$?

As is well-known, ZFC proves the equipotency of $\mathbb{R}$ and $\mathcal{P}(\mathbb{Q}).$ Is there a nice characterization of those linearly ordered sets $L$ which, like $\mathbb{Q}$, have the property that their completion by Dedekind-cuts $\mathcal{C}(L)$ is equipotent with their powerset $\mathcal{P}(L)$?

Edit. At the risk of introducing a non-standard concept into a perfectly standard question, I am especially interested in the case where $L$ is unbounded and "locally homogeneous," in the sense that for all $x,y \in L$ and all $x',y' \in L,$ we have that if $x<y$ and $x'<y'$, then the interval $(x,y)$ is isomorphic to the interval $(x',y')$. (The purpose of this condition is just to cut out all the crazy mish-mash orders involving mixtures of scattered and dense linear orders with complete and incomplete segments all smashed together into some crazy pattern. See Arthur's answer here for an example of a homogeneous order different from $\mathbb{R}$ and $\mathbb{Q}$).

As we all know, ZFC proves the equipotency of $\mathbb{R}$ and $\mathcal{P}(\mathbb{Q}).$ Is there a nice characterization of those linearly ordered sets $L$ which, like $\mathbb{Q}$, have the property that their completion by Dedekind-cuts $\mathcal{C}(L)$ is equipotent with their powerset $\mathcal{P}(L)$?

As is well-known, ZFC proves the equipotency of $\mathbb{R}$ and $\mathcal{P}(\mathbb{Q}).$ Is there a nice characterization of those linearly ordered sets $L$ which, like $\mathbb{Q}$, have the property that their completion by Dedekind-cuts $\mathcal{C}(L)$ is equipotent with their powerset $\mathcal{P}(L)$?