In this posting what was termed as "Proper Class Choice" principle turned to be equivalent to Global Choice over the base theory of "MK-Foundation -Limitation of size + Set Replacement*". However if the restriction to relations with set domains was placed (as in this posting), then it became equivalent to "set choice + collection" over the above base theory. This made me think of the following restriction:

Axiom of Small Class Choice :$$\forall \ relation \ R^{ |dom| < |V|} \exists F \subset R \ [F: dom(R) \to rng(R)]$$

In English: for every relation from a small class domain, there is a subclass of it that is a function from the same domain.

A class is said to be small if and only if it is strictly smaller in cardinality than the universe $V$ of all sets.

$^*$ Set Replacement is the assertion: "replacement classes from sets, are sets".

Questions

 

1. Are there models of the "base theory + small class choice" in which there exist small proper classes?

2. If yes, is small class choice principle stronger than "choice over sets + collection" over the base theory? I mean are there models satisfying the base theory in which choice over sets + collection is satisfied but small class choice fails?

In this posting what was termed as "Proper Class Choice" principle turned to be equivalent to Global Choice over the base theory of "MK-Foundation -Limitation of size + Set Replacement*". However if the restriction to relations with set domains was placed (as in this posting), then it became equivalent to "set choice + collection" over the above base theory. This made me think of the following restriction:

Axiom of Small Class Choice :$$\forall \ relation \ R^{ |dom| < |V|} \exists F \subset R \ [F: dom(R) \to rng(R)]$$

In English: for every relation from a small class domain, there is a subclass of it that is a function from the same domain.

A class is said to be small if and only if it is strictly smaller in cardinality than the universe $V$ of all sets.

$^*$ Set Replacement is the assertion: "replacement classes from sets, are sets".

Questions

1. Are there models of the "base theory + small class choice" in which there exist small proper classes?

2. If yes, is small class choice principle stronger than "choice over sets + collection" over the base theory? I mean are there models satisfying the base theory in which choice over sets + collection is satisfied but small class choice fails?

In this posting what was termed as "Proper Class Choice" principle turned to be equivalent to Global Choice over the base theory of "MK-Foundation -Limitation of size + Set Replacement*". However if the restriction to relations with set domains was placed (as in this posting), then it became equivalent to "set choice + collection" over the above base theory. This made me think of the following restriction:

Axiom of Small Class Choice :$$\forall \ relation \ R^{ |dom| < |V|} \exists F \subset R \ [F: dom(R) \to rng(R)]$$

In English: for every relation from a small class domain, there is a subclass of it that is a function from the same domain.

A class is said to be small if and only if it is strictly smaller in cardinality than the universe $V$ of all sets.

$^*$ Set Replacement is the assertion: "replacement classes from sets, are sets".

Questions

1. Are there models of the "base theory + small class choice" in which there exist small proper classes?

2. If yes, is small class choice principle stronger than "choice over sets + collection" over the base theory?

In this posting what was termed as "Proper Class Choice" principle turned to be equivalent to Global Choice over the base theory of "MK-Foundation -Limitation of size + Set Replacement*". However if the restriction to relations with set domains was placed (as in this posting), then it became equivalent to "set choice + collection" over the above base theory. This made me think of the following restriction:

Axiom of Small Class Choice :$$\forall \ relation \ R^{ |dom| < |V|} \exists F \subset R \ [F: dom(R) \to rng(R)]$$

In English: for every relation from a small class domain, there is a subclass of it that is a function from the same domain.

A class is said to be small if and only if it is strictly smaller in cardinality than the universe $V$ of all sets.

$^*$ Set Replacement is the assertion: "replacement classes from sets, are sets".

Questions

1. Are there models of the "base theory + small class choice" in which there exist small proper classes?

2. If yes, is small class choice principle stronger than "choice over sets + collection" over the base theory? I mean are there models satisfying the base theory in which choice over sets + collection is satisfied but small class choice fails?