User:Inconsistency/Introduction to Metamathematics

Axiom of Choice
This is the most controversial axiom of the bunch. It has nevertheless been shown that, if Set Theory is consistent without the Axiom of Choice, then it is also consistent with it. Various alternate theories have been formulated where the axiom of choice is replaced by some weaker statement (which would be a consequence of the full-fledged Axiom of Choice).

A proof using the Axiom of Choice  (or, possibly, one of its weaker counterparts) is usually called nonconstructive, and is rejected by constructivists. However, such a proof may serve to demonstrate [even to constructivists] that the negation of a nonconstructive theorem cannot possibly be shown to be true within the framework of the rest of Set Theory (since the Axiom of Choice must be consistent with it)...

The Axiom of Restriction Ruling out infinite descending membership chains. In 1917, Dmitri Mirimanov  discovered that certain models compatible with the axioms of ZF set theory allow the existence of infinite descending membership chains:  xn+1Îxn

John von Neumann found this repugnant and proposed his  Axiom of Restriction  to rule that out.

Philosophically, the axiom of restriction disallows sets beyond those created in finitely many applications of the other axioms of ZF.

Interestingly, if the axiom of dependent choice is postulated instead of the full axiom of choice, then the axiom of restriction implies regularity (so regularity no longer needs to be postulated as an axiom).