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A Note on Singular Cardinals in Set Theory Without Choice

We discuss how singular can cardinals be in absence of the axiom of choice. We show that, contrasting with known negative consistency results (of Gitik and others), certain positive results are provable. Then we pose some problems.

Equivalently, ℵ(n) is the least α such that on a set of size n there is no well-ordering of length α, ℵ * (n) is the least α such that on a set of size n there is no pre-well-ordering of length α.

Results on Singularity
Notations: Cov(l, m, n) means "A set of size n can be covered by m sets of size l ".
Cov(< l, m, n) and Cov(L, m, n) (where L is a class of cardinals) have the appropriate meanings.

Definition.
A cardinal n is singular iff Cov(< n, < n, n), and regular otherwise.
What is under AC?
Corollary. Assume AC. Then all the successor alephs are regular.
What happens without AC?
Moreover, under a large cardinal hypothesis, so can be all uncountable alephs: Theorem (Gitik). All uncountable alephs can be singular.
Remark. What is the consistency strength?
Without successive singular alephs: The same as of ZFC.
So, in general case: A proper class of Woodins.
is consistent (modulo large cardinals).
(Really, their technique gives slightly more.) In general, the problem remains open.
Question: How singular can cardinals be without AC? in the following sense: How small are l ≤ n and m ≤ n satisfying (i) Cov(< l, < n, n)?
On (iii): Specker's problem is a partial case.

On (ii):
The answer is As small as possible since Gitik's model satisfies Cov(< n, ℵ 0 , n) for all (not only well-ordered) n.

On (i):
For well-ordered n, the answer is l < n is impossible.
Next question: Let Cov(l, m, n), is n estimated via l and m? (when n is not well-ordered). Without Foundation, the answer is No Even in the simplest case l = 2 and m = ℵ 0 such an estimation of n is not provable: Theorem 2. It is consistent that for any p there exists n p such that Cov(2, ℵ 0 , n).
The proof uses a generalization of permutation model technique to the case of a proper class of atoms. We use non-well-founded sets instead of atoms.
Since Cov(n, n, 2 n ) is consistent, the result is near optimal.
Another corollary is that Specker's request, even in a weaker form, gives the least possible evaluation of ℵ * (2 λ ) (which is λ ++ ): So, if there exists a model which gives the positive answer to Specker's problem, then in it, all the cardinals ℵ * (2 λ ) have the least possible values.
As the last corollary, we provide a "pathology" when a set admits neither well-ordered covering (of arbitrary size) by sets of smaller size, nor covering of smaller size by well-orderable sets (of arbitrary size). Moreover, it can be the real line: Perhaps if F makes no successive cardinals singular, it is rather easy; otherwise very hard.