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Global choice is equivalent to saying $|V|=ON$, while ordinary choice is equivalent to saying that $V= H_{< ON}$. So the relative cardinalities of $V$ and $ON$ seems to affect the degree of choice available over classes. Now can this be affected by how much the cardinality of $ON$ is far from that of $V$?

I mean suppose we work in $\sf MK$ and axiomatize that $V=H_{<ON}$ [in other words all sets are injective to $ON$], and that $|X| \not \leq |ON| \leftrightarrow |X|=|V|$, then this says that the cardinality of the class of all set ordinals is immediately below that of the class of all sets, and that there are no other cardinalities other than those of sets, so the cardinalities of $V$ and $ON$ are very near.

On the other hand suppose instead of the above biconditional we stated that there are $|V|$ (or even strictly more than $|V|$) many proper class cardinalities$^\dagger$ other than $|ON|$ and $|V|$, then this would entail that the cardinality of the class of all set ordinals is quite distant from that of $V$ if not extremely lower.

Would those have different choice properties, that in some sense form a spectrum intermediate between set choice and global choice? For example, the former entailing more amount of choice than the latter?

$^\dagger$ this can be captured in $\sf MK$ by saying that there is a class $U$ that is the disjoint union of pairwise non-equinumerous proper classes such that the domain class of $U$ is equinumerous with $V$ [more clearly $U$ is a class of ordered pairs, where it's domain (the class of all first projections of elements of $U$) is equinumerous with $V$, and for each element $k$ in the domain of $U$, the class $K$ of all second projections of elements of $U$ whose first projection is $k$, is a proper class, we shall call such class $K$ as a proper class element of $U$, and if $K_1, K_2$ are proper class elements of $U$ then they are not equinumerous]. If we further demand that any two proper class elements of $U$ have comparable cardinalities, then $ON$ would be hugely lower in cardinality than $V$.

We may even go further along that method by adding that no such union class $U$ can satisfy that for every proper class $X$ there is a proper class element of $U$ that is equinumerous to $X$; in other words there are more than $|V|$ many non-equinumerouse proper classes!

My question is not just about the consistency of those situations, which I think it's there, but more importantly about the use of those situations as choice properties over classes, do they have any known use in set\class theories as intermediate kinds of choice between set choice and global choice, since those would be so.

Global choice is equivalent to saying $|V|=ON$, while ordinary choice is equivalent to saying that $V= H_{< ON}$. So the relative cardinalities of $V$ and $ON$ seems to affect the degree of choice available over classes. Now can this be affected by how much the cardinality of $ON$ is far from that of $V$?

I mean suppose we work in $\sf MK$ and axiomatize that $V=H_{<ON}$ [in other words all sets are injective to $ON$], and that $|X| \not \leq |ON| \leftrightarrow |X|=|V|$, then this says that the cardinality of the class of all set ordinals is immediately below that of the class of all sets, and that there are no other cardinalities other than those of sets, so the cardinalities of $V$ and $ON$ are very near.

On the other hand suppose instead of the above biconditional we stated that there are $|V|$ (or even strictly more than $|V|$) many proper class cardinalities$^\dagger$, then this would entail that the cardinality of the class of all set ordinals is quite distant from that of $V$ if not extremely lower.

Would those have different choice properties, that in some sense form a spectrum intermediate between set choice and global choice? For example, the former entailing more amount of choice than the latter?

$^\dagger$ this can be captured in $\sf MK$ by saying that there is a class $U$ that is the disjoint union of pairwise non-equinumerous proper classes such that the domain class of $U$ is equinumerous with $V$ [more clearly $U$ is a class of ordered pairs, where it's domain (the class of all first projections of elements of $U$) is equinumerous with $V$, and for each element $k$ in the domain of $U$, the class $K$ of all second projections of elements of $U$ whose first projection is $k$, is a proper class, we shall call such class $K$ as a proper class element of $U$, and if $K_1, K_2$ are distinct proper class elements of $U$ then they are not equinumerous]. If we further demand that any two proper class elements of $U$ have comparable cardinalities, then $ON$ would be hugely lower in cardinality than $V$.

We may even go further along that method by adding that no such union class $U$ can satisfy that for every proper class $X$ there is a proper class element of $U$ that is equinumerous to $X$; in other words there are more than $|V|$ many non-equinumerouse proper classes!

My question is not just about the consistency of those situations, which I think it's there, but more importantly about the use of those situations as choice properties over classes, do they have any known use in set\class theories as intermediate kinds of choice between set choice and global choice, since those would be so.

Global choice is equivalent to saying $|V|=ON$, while ordinary choice is equivalent to saying that $V= H_{< ON}$. So the relative cardinalities of $V$ and $ON$ seems to affect the degree of choice available over classes. Now can this be affected by how much the cardinality of $ON$ is far from that of $V$?

I mean suppose we work in $\sf MK$ and axiomatize that $V=H_{<ON}$ [in other words all sets are injective to $ON$], and that $|X| \not \leq |ON| \leftrightarrow |X|=|V|$, then this says that the cardinality of the class of all set ordinals is immediately below that of the class of all sets, and that there are no other cardinalities other than those of sets, so the cardinalities of $V$ and $ON$ are very near.

On the other hand suppose instead of the above biconditional we stated that there are $|V|$ many proper class cardinalities$^\dagger$ other than $|ON|$ and $|V|$, then this would entail that the cardinality of the class of all set ordinals is quite distant from that of $V$ if not extremely lower.

Would those have different choice properties, that in some sense form a spectrum intermediate between set choice and global choice? For example, the former entailing more amount of choice than the latter?

$^\dagger$ this can be captured in $\sf MK$ by saying that there is a class $U$ that is the disjoint union of pairwise non-equinumerous proper classes such that the domain class of $U$ is equinumerous with $V$ [more clearly $U$ is a class of ordered pairs, where it's domain (the class of all first projections of elements of $U$) is equinumerous with $V$ and for each element $k$ in the domain of $U$, the class $K$ of all second projections of elements of $U$ whose first projection is $k$, is a proper class, we shall call such class $K$ as the proper class element of $U$]. If we further demand that any two proper classe elements of $U$ have comparable cardinalities, then $ON$ would be hugely lower in cardinality than $V$.

We may even go further along that method by adding that no such union class $U$ can satisfy that for every proper class $X$ there is a proper class element of $U$ that is equinumerous to $X$; in other words there are more than $|V|$ many non-equinumerouse proper classes!

My question is not just about the consistency of those situations, which I think it's there, but more importantly about the use of those situations as choice properties over classes, do they have any known use in set\class theories as intermediate kinds of choice between set choice and global choice, since those would be so.

Global choice is equivalent to saying $|V|=ON$, while ordinary choice is equivalent to saying that $V= H_{< ON}$. So the relative cardinalities of $V$ and $ON$ seems to affect the degree of choice available over classes. Now can this be affected by how much the cardinality of $ON$ is far from that of $V$?

I mean suppose we work in $\sf MK$ and axiomatize that $V=H_{<ON}$ [in other words all sets are injective to $ON$], and that $|X| \not \leq |ON| \leftrightarrow |X|=|V|$, then this says that the cardinality of the class of all set ordinals is immediately below that of the class of all sets, and that there are no other cardinalities other than those of sets, so the cardinalities of $V$ and $ON$ are very near.

On the other hand suppose instead of the above biconditional we stated that there are $|V|$ (or even strictly more than $|V|$) many proper class cardinalities$^\dagger$ other than $|ON|$ and $|V|$, then this would entail that the cardinality of the class of all set ordinals is quite distant from that of $V$ if not extremely lower.

Would those have different choice properties, that in some sense form a spectrum intermediate between set choice and global choice? For example, the former entailing more amount of choice than the latter?

$^\dagger$ this can be captured in $\sf MK$ by saying that there is a class $U$ that is the disjoint union of pairwise non-equinumerous proper classes such that the domain class of $U$ is equinumerous with $V$ [more clearly $U$ is a class of ordered pairs, where it's domain (the class of all first projections of elements of $U$) is equinumerous with $V$, and for each element $k$ in the domain of $U$, the class $K$ of all second projections of elements of $U$ whose first projection is $k$, is a proper class, we shall call such class $K$ as a proper class element of $U$, and if $K_1, K_2$ are proper class elements of $U$ then they are not equinumerous]. If we further demand that any two proper class elements of $U$ have comparable cardinalities, then $ON$ would be hugely lower in cardinality than $V$.

We may even go further along that method by adding that no such union class $U$ can satisfy that for every proper class $X$ there is a proper class element of $U$ that is equinumerous to $X$; in other words there are more than $|V|$ many non-equinumerouse proper classes!

My question is not just about the consistency of those situations, which I think it's there, but more importantly about the use of those situations as choice properties over classes, do they have any known use in set\class theories as intermediate kinds of choice between set choice and global choice, since those would be so.

Global choice is equivalent to saying $|V|=ON$, while ordinary choice is equivalent to saying that $V= H_{< ON}$. So the relative cardinalities of $V$ and $ON$ seems to affect the degree of choice available. Now can this be affected by how much the cardinality of $ON$ is far from that of $V$?

I mean suppose we axiomatize that $V=H_{<ON}$ and that $|X| \not \leq |ON| \leftrightarrow |X|=|V|$, then this says that the cardinality of the class of all set ordinals is immediately below that of the class of all sets, and that there are no other cardinalities other than those of sets, so the cardinalities of $V$ and $ON$ are very near.

On the other hand suppose instead of the above biconditional we stated that there are $|V|$ many proper class cardinalities$^\dagger$ other than $ON$ and $V$, then this would entail that the cardinality of the class of all set ordinals is quite distant from that of $V$ if not extremely lower.

Would those have different choice properties, that in some sense form a spectrum intermediate between set choice and global choice? For example, the former entailing more amount of choice than the latter?

$^\dagger$ this can be captured by saying that there is a class $U$ that is the disjoint union of pairwise non-equinumerous proper classes such that the domain class of $U$ is equinumerous with $V$. If we further demand that any two of those proper classes have comparable cardinalities, then $ON$ would be hugely lower in cardinality than $V$.

We may even go further along that method by adding that no such union class $U$ can satisfy that for every proper class $X$ there is a proper class element of $U$ (i.e. a class of all images of a single element in $\operatorname {dom}(U)$, under $U$) that is equinumerous to $X$, in other words there are more than $|V|$ many non-equinumerouse proper classes!

Global choice is equivalent to saying $|V|=ON$, while ordinary choice is equivalent to saying that $V= H_{< ON}$. So the relative cardinalities of $V$ and $ON$ seems to affect the degree of choice available over classes. Now can this be affected by how much the cardinality of $ON$ is far from that of $V$?

I mean suppose we work in $\sf MK$ and axiomatize that $V=H_{<ON}$ [in other words all sets are injective to $ON$], and that $|X| \not \leq |ON| \leftrightarrow |X|=|V|$, then this says that the cardinality of the class of all set ordinals is immediately below that of the class of all sets, and that there are no other cardinalities other than those of sets, so the cardinalities of $V$ and $ON$ are very near.

On the other hand suppose instead of the above biconditional we stated that there are $|V|$ many proper class cardinalities$^\dagger$ other than $|ON|$ and $|V|$, then this would entail that the cardinality of the class of all set ordinals is quite distant from that of $V$ if not extremely lower.

Would those have different choice properties, that in some sense form a spectrum intermediate between set choice and global choice? For example, the former entailing more amount of choice than the latter?

$^\dagger$ this can be captured in $\sf MK$ by saying that there is a class $U$ that is the disjoint union of pairwise non-equinumerous proper classes such that the domain class of $U$ is equinumerous with $V$ [more clearly $U$ is a class of ordered pairs, where it's domain (the class of all first projections of elements of $U$) is equinumerous with $V$ and for each element $k$ in the domain of $U$, the class $K$ of all second projections of elements of $U$ whose first projection is $k$, is a proper class, we shall call such class $K$ as the proper class element of $U$]. If we further demand that any two proper classe elements of $U$ have comparable cardinalities, then $ON$ would be hugely lower in cardinality than $V$.

We may even go further along that method by adding that no such union class $U$ can satisfy that for every proper class $X$ there is a proper class element of $U$ that is equinumerous to $X$; in other words there are more than $|V|$ many non-equinumerouse proper classes!

My question is not just about the consistency of those situations, which I think it's there, but more importantly about the use of those situations as choice properties over classes, do they have any known use in set\class theories as intermediate kinds of choice between set choice and global choice, since those would be so.