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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeAug 19th 2021
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexAdded: *** ## Cofinality without choice Assuming the consistency of the existence of arbitrarily large [[strongly compact cardinals]], it is consistent with ZF that every infinite set is a countable union of sets of smaller cardinality. See \cite{Gitik}. ## Related concepts * [[initial functor]], [[final functor]] ## References * M. Gitik, _All uncountable cardinals can be singular_, Israel Journal of Mathematics 35:1-2 (1980), 61-88. [doi](http://dx.doi.org/10.1007/bf02760939). *** <a href="https://ncatlab.org/nlab/revision/diff/cofinality/5">diff</a>, <a href="https://ncatlab.org/nlab/revision/cofinality/5">v5</a>, <a href="https://ncatlab.org/nlab/show/cofinality">current</a>

   Added:

   ---

   Cofinality without choice

   Assuming the consistency of the existence of arbitrarily large strongly compact cardinals, it is consistent with ZF that every infinite set is a countable union of sets of smaller cardinality. See \cite{Gitik}.

   Related concepts

   • initial functor, final functor

   References

   • M. Gitik, All uncountable cardinals can be singular, Israel Journal of Mathematics 35:1-2 (1980), 61-88. doi.

   ---

   diff, v5, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeAug 19th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have expanded out ([here](https://ncatlab.org/nlab/show/cofinality#CofinalityWithoutChoice)) your line > it is consistent with ZF that to: > it is consistent with Zermelo-Fraenkel-[[set theory]] without the [[axiom of choice]] that If that's not what is meant, please expand to clarify. <a href="https://ncatlab.org/nlab/revision/diff/cofinality/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/cofinality/6">v6</a>, <a href="https://ncatlab.org/nlab/show/cofinality">current</a>

   I have expanded out (here) your line

   it is consistent with ZF that

   to:

   it is consistent with Zermelo-Fraenkel-set theory without the axiom of choice that

   If that’s not what is meant, please expand to clarify.

   diff, v6, current