Fundamental sequence (set theory)

Given an ordinal \alpha, a fundamental sequence for \alpha is a sequence (\alpha[n])_{n\in\mathbb N} such that \forall(n\in\mathbb N)(\alpha[n] and \textrm{sup}\{\alpha[n]\mid n\in\mathbb N\}=\alpha. ==Examples==

The following is a common assignment of fundamental sequences to all limit ordinals less than \varepsilon_0. • \omega^{\alpha+1}[n]=\omega^\alpha\cdot(n+1) • \omega^\alpha[n]=\omega^{\alpha[n]} for limit ordinals \alpha • (\omega^{\alpha_1}+\ldots+\omega^{\alpha_k})[n]=\omega^{\alpha_1}+\ldots+(\omega^{\alpha_k}[n]) for \alpha_1 \geq \dots \geq \alpha_k. This is very similar to the system used in the Wainer hierarchy. ==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 \phi_\alpha 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 \omega_1. 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. Additional conditions A system of fundamental sequences up to \alpha is said to have the Bachmann property if for all ordinals \alpha,\beta in the domain of the system and for all n\in\mathbb N, \alpha[n]. If a system of fundamental sequences has the Bachmann property, all the functions in its associated fast-growing hierarchy are monotone, and f_\beta eventually dominates f_\alpha when \alpha. ==References==