Ordinal notation system

An ordinal notation system (also called an ordinal notation informally) is a system of names for ordinals, or equivalently, a well-ordered set of objects that "can be considered names". There may not be a formal restriction of what objects can be considered names, but most common examples include strings, sequences of other such objects, trees/hydras, and terms built up from constants and functions. Most authors also require the well-order to be recursive, and this requirement is included in the definition of proof-theoretic ordinals that uses ordinal notation systems.

Notable ordinal notation systems include:

- Cantor normal form

- Primitive sequence system

- Pair sequence system

- Ordinal notation systems associated to ordinal collapsing functions

- Taranovsky's ordinal notations (the ones that are well-ordered)

- Patterns of resemblance

- Bashicu matrix system

- Y sequence (as long as it is well-ordered)

As can be seen in this list, proposed ordinal notation systems need to be proven well-ordered in order to be considered ordinal notation systems with certainty, and there are notable cases where this is not proven yet.