John von Neumann's ordinal numbers

John von Neumann’s ordinal numbers are a way to rigorously set up a system for understanding "ordering" or "sequence" in mathematics. He defined each ordinal number as a set containing all earlier ordinals, which allows a clear hierarchy of position or order—like labels for positions in an endless list. The first ordinal is empty (no elements), the next contains just the first, and so forth, building an infinite hierarchy. This framework helps mathematicians study concepts of size, order, and infinity systematically, forming a foundation for set theory and much of modern mathematics.