This question is a follow up to that posting.

Recall the definition of super\hyper\ultra-singular set given in the linked posting.

Is there a model of $\sf ZF$ in which every uncountable set is super-singular?

Same question but in terms of the other two kinds of singularity?

We know that a Gitik model of $\sf ZF$ has every uncountable set being singular? But can we have Gitik models satisfying any of the above three conditions?

To re-iterate the definitions:

A set $A$ is super-singular, if and only if, there exists a set $x$ such that $|A|=|\bigcup x|$ and $| A| > |x|$, and for each $y \in x$ we have: $ |y| \not > |x|$ .

A set $A$ is hyper-singular, if and only if, there exists a set $x$ such that: $|A|=|\bigcup x|$ and $| A| > |x|$, and for each $y \in x$ we have: $|y| \not > |x| \land |y| \neq |x|$.

A set $A$ is ultra-singular, if and only if, there exists a set $x$ such that $|A|=|\bigcup x|$ and $| A| > |x|$, and for each $y \in x$ we have: $|y| < |x|$.

This question is a follow up to that posting.

Recall the definition of super/hyper/ultra-singular set given in the linked posting.

Is there a model of $\sf ZF$ in which every uncountable set is super-singular?

Same question but in terms of the other two kinds of singularity?

We know that a Gitik model of $\sf ZF$ has every uncountable set being singular? But can we have Gitik models satisfying any of the above three conditions?

To re-iterate the definitions:

A set $A$ is super-singular, if and only if, there exists a set $x$ such that $|A|=|\bigcup x|$ and $| A| > |x|$, and for each $y \in x$ we have: $ |y| \not > |x|$ .

A set $A$ is hyper-singular, if and only if, there exists a set $x$ such that: $|A|=|\bigcup x|$ and $| A| > |x|$, and for each $y \in x$ we have: $|y| \not > |x| \land |y| \neq |x|$.

A set $A$ is ultra-singular, if and only if, there exists a set $x$ such that $|A|=|\bigcup x|$ and $| A| > |x|$, and for each $y \in x$ we have: $|y| < |x|$.

This question is a follow up to that posting.

Recall the definition of super\hyper\ultra-singular set given in the linked posting.

Is there a model of $\sf ZF$ in which every uncountable set is super-singular?

Same question but in terms of the other two kinds of singularity?

We know that a Gitik model of $\sf ZF$ has every uncountable set being singular? But can we have Gitik models satisfying any of the above three conditions?

This question is a follow up to that posting.

Recall the definition of super\hyper\ultra-singular set given in the linked posting.

Is there a model of $\sf ZF$ in which every uncountable set is super-singular?

Same question but in terms of the other two kinds of singularity?

We know that a Gitik model of $\sf ZF$ has every uncountable set being singular? But can we have Gitik models satisfying any of the above three conditions?

To re-iterate the definitions:

A set $A$ is super-singular, if and only if, there exists a set $x$ such that $|A|=|\bigcup x|$ and $| A| > |x|$, and for each $y \in x$ we have: $ |y| \not > |x|$ .

A set $A$ is hyper-singular, if and only if, there exists a set $x$ such that: $|A|=|\bigcup x|$ and $| A| > |x|$, and for each $y \in x$ we have: $|y| \not > |x| \land |y| \neq |x|$.

A set $A$ is ultra-singular, if and only if, there exists a set $x$ such that $|A|=|\bigcup x|$ and $| A| > |x|$, and for each $y \in x$ we have: $|y| < |x|$.