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 $\mathrm{\omega}_1$. The $$n^{\text{th}}$$ element of the fundamental sequence of $$\alpha$$ is commonly denoted $$\alpha[n]$$, although it may be denoted $$\alpha_n$$ or $$\{\alpha\}(n)$$. Additionally, some authors may allow fundamental sequences to be defined on successor ordinals. The term dates back to (at the latest) Veblen's construction of normal functions $$\varphi_\alpha$$, while the concept dates back to Hardy's 1904 attempt to construct a set of cardinality $$\aleph_1$$.

Definition
Given an ordinal $$\alpha$$, a fundamental sequence for $$\alpha$$ is a sequence $$(\alpha[n])_{n\in\mathbb N}$$ such that $$\forall(n\in\mathbb N)(\alpha[n]<\alpha)$$ and $$\textrm{sup}\{\alpha[n]\mid n\in\mathbb N\}=\alpha$$. 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 $$\varepsilon_0$$. This is very similar to the system used in the Wainer hierarchy.
 * $$\omega^{\alpha+1}[n]=\omega^\alpha\cdot(n+1)$$
 * $$\omega^\alpha[n]=\omega^{\alpha[n]}$$ for limit ordinals $$\alpha$$
 * $$(\omega^{\alpha_1}+\ldots+\omega^{\alpha_k})[n]=\omega^{\alpha_1}+\ldots+(\omega^{\alpha_k}[n])$$ for $$\alpha_1 \geq \dots \geq \alpha_k$$.

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 $$n\in\mathbb N$$, $$\alpha[n]<\beta<\alpha\implies\alpha[n]<\beta[0]$$. 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 $$\alpha<\beta$$.