The Regular Extension Axiom is a foundational axiom which asserts the existence of arbitrarily large regular cardinal-like sets. It has several variants, some of which are provable in ZF, some of which are provable from the axiom of choice or weaker variants thereof such as SVC, and some of which are not even provable in ZFC.

Variants

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References

Michael Rathjen? and Robert Lubarsky?, On the regular extension axiom and its variants, PDF – broken link