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(note: in what follows by "consistent" I mean "consistent relative to large cardinals")

My question regards the exact statement of result which Gitik has proven in his paper "All Uncountable Cardinals Can Be Singular". In the abstract, he claims to have procen consistency of the following:

Every infinite set is a countable union of sets of smaller cardinality.

However, theorem I talks about consistency of the following:

For all $\alpha$ cofinality of $\aleph_\alpha$ is $\aleph_0$.

Since we don't have choice in our hands, these two formulations are not necessarily equivalent, because of non-well-orderable cardinals. Indeed, we can't even have countable choice here, so we might have to deal with infinite Dedekind-finite cardinalities. This lead me to asking this question:

Has the consistency of

Every infinite set is a countable union of sets of smaller cardinality.

actually been proven? How about

Every infinite set is a countable union of nonempty sets of smaller cardinality.

I believe the latter is equivalent to the former +"all Dedekind finite sets are finite".

Thanks in advance.

Edit: as Arthur Fisher points out, part (a) of theorem II of the same paper answers exactly the first part of my question. Second part however still stands. I suspect Gitik's model doesn't have any infinite Dedekind-finite sets, but the paper is far beyond my understanding.

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  • $\begingroup$ How do you define "singular" for non-aleph cardinals? This is a very nontrivial thing. $\endgroup$ – Asaf Karagila Dec 20 '15 at 12:44
  • $\begingroup$ @AsafKaragila I don't really need the notion of "singular" per se for my question, since in the body I talk about cardinals being sums of countably many smaller sets. $\endgroup$ – Wojowu Dec 20 '15 at 12:45
  • $\begingroup$ See "additional remarks" in my answer here: math.stackexchange.com/questions/775565/… $\endgroup$ – Asaf Karagila Dec 20 '15 at 12:47
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In the final part of his paper

M. Gitik, All uncountable cardinals can be singular, Israel J. Math. 35 (1980), no. 1-2, 61--88.

Gitik proves that in his model every set is the countable union of sets with a smaller cardinality, and moreover if we close all the countable sets under countable unions, we obtain every set.

He then proves that it is consistent that every ordinal has countable cofinality, but there is a set which is not the countable union of smaller sets.

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