User blog comment:Ubersketch/Ordinals with transfinite FS expansions/@comment-35470197-20190701222433/@comment-32213734-20190702125139

In Axiom of regularity Wikipedia article it is said: "Given the other axioms of Zermelo–Fraenkel set theory, the axiom of regularity is equivalent to the axiom of induction".

In Transfinite induction Wikipedia article it is said: "Let P(α) be a property defined for all ordinals α. Suppose that whenever P(β) is true for all β < α, then P(α) is also true. Then transfinite induction tells us that P is true for all ordinals".

So, let P(α) is "α has no infinite descending chain". Let
 * α > β > β1 > β2 > ...

Let P(β) is true for any such β. Then descending chain
 * β > β1 > β2 > ...

is finite. Hence
 * α > β > β1 > β2 > ...

is also finite. So, P(α) is true. From transfinite induction it follows that P(α) is true for any ordinal α, that is ordinals with infinite descending chain do not exist.

Also, yes, it can be proved using axiom of regularity. In Axiom of regularity Wikipedia article it is said: "The axiom implies that no set is an element of itself, and that there is no infinite sequence (an) such that ai+1 is an element of ai for all i." Since an ordinal is set of all smaller ordinals, then in any descending chain of ordinals
 * α0 > α1 > α2 > ...

any αi+1 is element of αi. So, from axiom of regularity it follows that there is no infinite descending chain of ordinals.