Generalization of Amorphous Sets We say a set $S$ is "$\aleph_{\alpha}$-amorphous" or "amorphous in $\aleph_{\alpha}$" if $\aleph_{\alpha}$ is the minimum $\aleph$ number such that $|S|\not<\aleph_{\alpha}$ and every $A\subseteq S$ satisfies either $|A|<\aleph_{\alpha}$ or $|S\setminus A|<\aleph_{\alpha}$.
This is a sort of generalization of amorphous sets I thought up based on the definition of an $\aleph_{1}$-amorphous set I saw (I don't quite remember where). Standard amorphous sets (i.e. ones in which every subset is either finite or co-finite) are $\aleph_{0}$-amorphous. I have a few questions:
Can these exist for all $\aleph_{\alpha}$ (in $\sf{ZF}$)? How about sets amorphous in well-orderable inaccessible cardinals?
Because $\aleph_{\alpha}\cdot\aleph_{\alpha}=\aleph_{\alpha}$, it seems that if $|S|\geq\aleph_{\alpha}$ then $S$ is certainly not amorphous in $\aleph_{\alpha}$. Thus, is it correct to say $\aleph_{\alpha}$-amorphous sets have Hartogs number $\aleph_{\alpha}$?
No $\aleph_{0}$-amorphous set can be linearly ordered. Is the same true for $\aleph_{\alpha}$-amorphous sets as well? It seems like a similar proof would work.
I am also interested in the comparability of the cardinalities of $\aleph_{\alpha}$-amorphous sets. For example, is it the case that any two $\aleph_{0}$-amorphous sets $A$ and $B$ have an injection from $A$ to $B$ or vice versa? How about $\aleph_{\alpha}$-amorphous sets?
Thanks.
 A: Yes, these sets can exist for all infinite well-ordered cardinals. The construction is essentially the same type. We can even have $\sf DC_{<\operatorname{cf}(\kappa)}$ along with a $\kappa$-amorphous set.
For $\aleph_1$-amorphous sets, those can be linearly ordered. In fact, there can be such a set of reals. 
The reason the usual proof doesn't work is that there are different linear orders which are uncountable but every proper initial segment is countable (e.g. $\omega_1$, or $\omega_1\times\Bbb Q$ or $\omega_1\times\Bbb Q\times\Bbb Z$, etc.) which means that the usual proof that relies on "$\Bbb N$ is the unique linear order without a maximal element in which every proper initial segment is finite" does not translate to a general one.
And as for comparability, we can arrange that for every $\kappa$ there is a proper class of pairwise incomparable $\kappa$-amorphous sets. How's that on for size? The idea is relatively simple: 
First consider a symmetric extension where we added a $\kappa$-amorphous set for some $\kappa$, let's say using Cohen subsets of some regular cardinal $\lambda$. 
Now fix a coding of pairs $(\xi,\zeta)$ of ordinals. In the $\alpha$th stage, if $\aleph_\alpha$ is a regular cardinal, take $\kappa=\aleph_\xi$ and $\lambda=\aleph_\alpha$, and this is our symmetric system. Now take an Easton support product of these forcings, and by genericity argument two amorphous sets coming from different symmetric systems must be incomparable. 
(What's clever is that you can use a finite support product and still preserve $\sf ZF$ in the outcome model, at least if you've chosen the obvious symmetric system in the first step above.)
