Fundamental sequence (set theory)
In set theory, a mathematical discipline, a fundamental sequence is a cofinal sequence of ordinals all below a given limit ordinal. Depending on author, fundamental sequences may be restricted to ω-sequences only or permit fundamental sequences of length [First uncountable ordinal|]. The element of the fundamental sequence of is commonly denoted, although it may be denoted or. Additionally, some authors may allow fundamental sequences to be defined on successor ordinals. The term dates back to Veblen's construction of normal functions, while the concept dates back to Hardy's 1904 attempt to construct a set of cardinality.
Definition
Given an ordinal, a fundamental sequence for is a sequence such that and. An additional restriction may be that the sequence of ordinals must be strictly increasing.Examples
The following is a common assignment of fundamental sequences to all limit ordinals less than [epsilon number|].- for limit ordinals
- for.
Usage
Fundamental sequences arise in some settings of definitions of large countable ordinals, definitions of hierarchies of fast-growing functions, and proof theory. Bachmann defined a hierarchy of functions in 1950, providing a system of names for ordinals up to what is now known as the Bachmann–Howard ordinal, by defining fundamental sequences for namable ordinals below. This system was subsequently simplified by Feferman and Aczel to reduce the reliance on fundamental sequences.The fast-growing hierarchy, Hardy hierarchy, and slow-growing hierarchy of functions are all defined via a chosen system of fundamental sequences up to a given ordinal. The fast-growing hierarchy is closely related to the Hardy hierarchy, which is used in proof theory along with the slow-growing hierarchy to majorize the provably computable functions of a given theory.