An amorphous set is an infinite set that cannot be partitioned into 2 infinite subsets. An amorphous cardinality is the cardinality of an amorphous set. Working in $\sf ZF$, it is consistent that amorphous sets exist.

Amorphous sets necessarily don't have a lot of structure. For example, for an amorphous set $A$, there can't be an infinite well-ordered chain of subsets of $A$, since the differences of consecutive elements in such a chain can be used to partition the set. However, if we are instead given a chain of **cardinalities** below $A$, $(\mathfrak c_i)_{i \in \mathbb N}$
such that $\mathfrak c_i < \mathfrak c_j < |A|$ for $i < j$, this argument does not work, because we can't uniformly choose representatives (and injections) for each cardinality.

To me, the existence of "long" chains of cardinalities below a set implies that it is "big". If there is such a sequence of length $\kappa$, then even if there isn't a literal subset of $A$ with cardinality $\kappa$, $A$ does in a sense contain at least $\kappa$ elements. I am interested in knowing whether amorphous sets can be arbitrarily "big" according to this intuition.

I have 3 questions of increasing strength:

A monotonic sequence of cardinalities of length $\alpha$ is a function $s : \alpha \to \sf Card$ such that $s(\alpha) < s(\beta)$ for $\alpha < \beta$.

Given an ordinal $\alpha$, is it consistent that there exists a monotonic sequence of amorphous cardinalities of length $\alpha$?

Is it consistent that for every ordinal $\alpha$ there exists a monotonic sequence of amorphous cardinalities of length $\alpha$?

Is it consistent that there is a monotonic class sequence $F : \sf Ord \to \sf Card$ of amorphous cardinalities?