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Let's work in a set theory without assuming AC (for instance, but not necessarily, ZF). Fix a set $k$ satisfying $k\times k \simeq k$, and consider its powerset $X = 2^k$. I have a technical condition that is satisfied whenever $X$ is well-orderable, but I can't tell what other examples there might be. Excuse the peculiar phrasing: I'm aiming for the most constructive means possible (by "well-orderable" I mean classically so: a total order such that every inhabited subset has a least element. This is equivalent to $X$ being a choice object).

Let $p\colon X \to I$ be a partition of $X$ into subsets $X_i=p^{-1}(i)\lt X$, $i\in I$ such that the images of all functions $f_i\colon X_i \to X$ have inhabited complement $\forall i \in I$. Assume I is minimal in the $\lt$-ordering of sets by cardinality (if $X$ is well-orderable, then $I$ is unique and $I\simeq cf(X)$). The condition is as follows:

For all partitions as just described, and all families of functions $(f_i\colon X_i \to X\mid i \in I)$, the surjection $$ \coprod_{i\in I} X_i \setminus f_i(X_i) \to I $$ has a section.

The consequence of this that I want is that I can choose a point in the complement of every map $X \to X^I$, namely $X < X^I$. For well-orderable $X$ this is a corollary of a restriction of König's theorem to powers of well-orderable sets in place of arbitrary products of families.

I would like to know if there are examples that aren't well-orderable, or otherwise how close to be able to well-order $X$ this gets. Note in particular, that we could take every function $f_i$ to be constant at a fixed point $x\in X$, and so this case implies that we should be able to take a section of $X\setminus\lbrace x\rbrace \to I$ and hence of $p$, for $p$ and $I$ as above.

Note that I don't quite have the result that $X$ is a choice object (equivalently, classically well-orderable). I'm not demanding the existence of sections for arbitrary surjections, or more generally for total relation into $X$ (a fine distinction, I'm sure).

Edit: $X$ being a choice object is equivalent to saying that there is a choice function from the set $P_+X$ of inhabited subsets to $X$. I am only asking for a choice function on the subset of $P_+X$ given by those subsets of $X$ of strictly smaller cardinality, call it $P^{\lt X}_+X$.

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  • $\begingroup$ I would also be happy assuming that $k$ is regular, in the sense that if given a partition $k to J$ with properties as above, and $J$ is minimal, then $k \simeq J$ $\endgroup$ – David Roberts Nov 27 '15 at 1:07
  • $\begingroup$ Do you want this to be true for all such $p:X\to I$, or only for some such $p:X\to I$? $\endgroup$ – Eric Wofsey Nov 27 '15 at 1:13
  • $\begingroup$ No, just the ones satisfying the conditions as given before the grey box. If it were for all $p$, then clearly I get a choice object. $\endgroup$ – David Roberts Nov 27 '15 at 1:14
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    $\begingroup$ Are you also assuming $X_i < X$? $\endgroup$ – François G. Dorais Nov 27 '15 at 13:53
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    $\begingroup$ @Joel, David: The blog post is up. I'd link it here, but there is a link to this post and I don't want to create a non-well founded chain of links. :-) $\endgroup$ – Asaf Karagila Nov 29 '15 at 21:08
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Prompted by the discussion in comments above, Asaf wrote a blog post Cofinality and the axiom of choice that gave the following example of a model of ZF:

It is consistent that every non well-orderable set has cofinality 2. This was shown by Monro (“Decomposable Cardinals“, Fund. Math. vol. 80 (1973), no. 2, 101–104.), more specifically the real numbers can have cofinality 2.

This destroys the inequality $\kappa < 2^{cf(\kappa)}$ resulting from König's theorem (I like to call this inequality König's corollary), for the notion of cofinality in the original question.

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