We consider four notions of strong inaccessibility that are equivalent in ZFC and show that they are not equivalent in ZF.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm194-2-3, author = {Andreas Blass and Ioanna M. Dimitriou and Benedikt L\"owe}, title = {Inaccessible cardinals without the axiom of choice}, journal = {Fundamenta Mathematicae}, volume = {193}, year = {2007}, pages = {179-189}, zbl = {1116.03045}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm194-2-3} }
Andreas Blass; Ioanna M. Dimitriou; Benedikt Löwe. Inaccessible cardinals without the axiom of choice. Fundamenta Mathematicae, Tome 193 (2007) pp. 179-189. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm194-2-3/