Working in $\sf ZF$, regarding an arbitrary Dedekind infinite set $x$, what's the relationship between the cardinality of $\mathcal P(x)$ and $\mathcal H_x$? The latter is the set of all sets hereditarily subnumerous to $x$, where subnumerous to means "is injective to" , by "$y$ is to hereditarily subnumerous to $x$" here it specifically means every element of the transitive closure of $\{y\} $ is subnumerous to $x$. Transitive closure defined in the customary way as the minimal transitive superset.

Are there special constraints on how they compare given Choice, GCH, V=L, etc.. Can they be incomparable? I'm interested in cases where $\mathcal H_x$ is subnumerous to $\mathcal P(x)$, whether strictly or not, especially outside of choice?

Also how matters would differ if we defined $\mathcal H_x$ as the set of all sets whose transitive closure is subnumerous to $x$? Especially as regards it being subnumerous to $\mathcal P(x) $ in absence of choice?

Working in $\sf ZF$, regarding an arbitrary Dedekind infinite set $x$, what's the relationship between the cardinality of $\mathcal P(x)$ and $\mathcal H^*_x$? The latter is the set of all sets hereditarily subnumerous to $x$, where subnumerous to means "is injective to" , by "$y$ is to hereditarily subnumerous to $x$" here it specifically means every element of the transitive closure of $\{y\} $ is subnumerous to $x$. Transitive closure defined in the customary way as the minimal transitive superset.

Are there special constraints on how they compare given Choice, GCH, V=L, etc.. Can they be incomparable? I'm interested in cases where $\mathcal H^*_x$ is subnumerous to $\mathcal P(x)$, whether strictly or not, especially outside of choice?

Also how matters would differ if we defined $\mathcal H^*_x$ as the set of all sets whose transitive closure is subnumerous to $x$? Especially as regards it being subnumerous to $\mathcal P(x) $ in absence of choice?

Working in $\sf ZF$, regarding an arbitrary Dedekind infinite set $x$, what's the relationship between the cardinality of $\mathcal P(x)$ and $\mathcal H_x$? The latter is the set of all sets hereditarily subnumerous to $x$, where subnumerous to means "is injective to" , by "$y$ is to hereditarily subnumerous to $x$" here it specifically means every element of the transitive closure of $\{y\} $ is subnumerous to $x$. Transitive closure defined in the customary way as the minimal transitive superset.

Are there special constraints on how they compare given Choice, GCH, V=L, etc.. Can they be incomparable? I'm interested in cases where $\mathcal H_x$ is subnumerous to $\mathcal P(x)$, whether strictly or not, especially outside of choice?

Working in $\sf ZF$, regarding an arbitrary Dedekind infinite set $x$, what's the relationship between the cardinality of $\mathcal P(x)$ and $\mathcal H_x$? The latter is the set of all sets hereditarily subnumerous to $x$, where subnumerous to means "is injective to" , by "$y$ is to hereditarily subnumerous to $x$" here it specifically means every element of the transitive closure of $\{y\} $ is subnumerous to $x$. Transitive closure defined in the customary way as the minimal transitive superset.

Are there special constraints on how they compare given Choice, GCH, V=L, etc.. Can they be incomparable? I'm interested in cases where $\mathcal H_x$ is subnumerous to $\mathcal P(x)$, whether strictly or not, especially outside of choice?

Also how matters would differ if we defined $\mathcal H_x$ as the set of all sets whose transitive closure is subnumerous to $x$? Especially as regards it being subnumerous to $\mathcal P(x) $ in absence of choice?

Working in $\sf ZF$, regarding an arbitrary Dedekind infinite set $x$, what's the relationship between the cardinality of $\mathcal P(x)$ and $\mathcal H_x$? The latter is the set of all sets hereditarily subnumerous to $x$. Are there special constraints on how they compare given Choice, GCH, V=L, etc.. Can they be incomparable? I'm interested in cases where $\mathcal H_x$ is subnumerous to $\mathcal P(x)$, whether strictly or not, especially outside of choice?

Working in $\sf ZF$, regarding an arbitrary Dedekind infinite set $x$, what's the relationship between the cardinality of $\mathcal P(x)$ and $\mathcal H_x$? The latter is the set of all sets hereditarily subnumerous to $x$, where subnumerous to means "is injective to" , by "$y$ is to hereditarily subnumerous to $x$" here it specifically means every element of the transitive closure of $\{y\} $ is subnumerous to $x$. Transitive closure defined in the customary way as the minimal transitive superset.

Are there special constraints on how they compare given Choice, GCH, V=L, etc.. Can they be incomparable? I'm interested in cases where $\mathcal H_x$ is subnumerous to $\mathcal P(x)$, whether strictly or not, especially outside of choice?