Are Ordinals totally ordered?

When delving into the realm of cryptography and its underlying mathematical foundations, a question that often arises is whether certain mathematical structures possess certain ordering properties. In this context, the concept of ordinals plays a crucial role. Ordinals are a generalization of the natural numbers that allow for a total ordering of sets. However, it begs the question: Are ordinals truly totally ordered? Do they satisfy the axioms of a total order, where any two elements are comparable and can be unambiguously placed in a sequence? This inquiry delves into the heart of the mathematical underpinnings of cryptography and the ways we categorize and structure data in this field.