Extensions
The Grzegorczyk hierarchy can be extended to transfinite ordinals. Such extensions define a fast-growing hierarchy. To do this, the generating functions must be recursively defined for limit ordinals (note they have already been recursively defined for successor ordinals by the relation ). If there is a standard way of defining a fundamental sequence, whose limit ordinal is, then the generating functions can be defined . However, this definition depends upon a standard way of defining the fundamental sequence. Rose (1984) suggests a standard way for all ordinals α < ε0.
The original extension was due to Martin Löb and Stan S. Wainer (1970) and is sometimes called the Löb–Wainer hierarchy.
Read more about this topic: Grzegorczyk Hierarchy
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