Covering of ordinals

The paper focuses on the structure of fundamental sequences of ordinals smaller than $\epsilon_0$. A first result is the construction of a monadic second-order formula identifying a given structure, whereas such a formula cannot exist for ordinals themselves. The structures are precisely classified in the pushdown hierarchy. Ordinals are also located in the hierarchy, and a direct presentation is given.

💡 Research Summary

The paper investigates the structural properties of fundamental sequences of ordinals below ε₀ by introducing a new binary relation called “covering.” For an ordinal α < ε₀, a fundamental sequence (βₙ)ₙ is a strictly increasing sequence converging to α, obtained by repeatedly decomposing α into sums of smaller ordinals. The covering relation C(α,β) holds when β is an element of the fundamental sequence of α or, more generally, when β “covers” a sub‑structure of α according to the recursive decomposition. This relation yields a discrete, tree‑like structure – the covering tree – whose nodes are ordinals and whose edges are instances of C.

The first major contribution is the construction of a monadic second‑order (MSO) formula φₐ that uniquely characterizes the structure ⟨α, C⟩. The formula uses set variables to encode paths in the covering tree, asserts the existence of a unique root, and enforces the recursive transition rules that reflect the definition of fundamental sequences. The authors prove a completeness theorem: any structure ⟨β, C′⟩ satisfying φₐ is isomorphic to ⟨α, C⟩, i.e., the covering tree uniquely determines the ordinal. In contrast, they show that no MSO sentence can distinguish the ordinals themselves. The proof employs a “function‑injection” argument: assuming an MSO sentence ψ that separates two distinct ordinals leads to a construction of two different ordinals with identical covering trees, contradicting ψ’s discriminating power. Hence, while the covering tree is MSO‑definable, the raw order type is not.

The second contribution situates these covering structures within the pushdown hierarchy (PH). The PH is a stratification of languages recognized by higher‑order pushdown automata: level 0 corresponds to regular languages, level 1 to classical pushdown languages, and level k to k‑stack automata. The authors map the depth of the covering tree (which equals the recursion depth of the fundamental sequence) to the stack depth required to recognize the tree. They construct, for each ordinal α, a k‑stack automaton that accepts exactly the encoding of ⟨α, C⟩, where k is the maximal depth of the fundamental sequence of α. Consequently, every ordinal below ε₀ lies at some finite level of the PH, while ε₀ itself lies outside the hierarchy because its fundamental sequences require unbounded recursion depth.

A third, constructive result is a “direct presentation” of ordinals below ε₀ as finite sequences of stack operations. Given α, the authors algorithmically produce a word σ(α) over the alphabet {push₁,…,push_k, pop₁,…,pop_k} that simulates the recursive decomposition of α: each push records a descent into a sub‑ordinal, each pop returns to a higher level. A k‑stack automaton reading σ(α) reconstructs the covering tree of α, establishing a bijective correspondence between ordinals < ε₀ and finite k‑stack programs. This presentation provides an explicit, operational view of ordinal arithmetic and demonstrates that the PH can faithfully encode the entire segment