An order‐theoretic characterization of the Schütte‐Veblen‐Hierarchy

Abstract

For f : On -+ On let supp(f) := {( : f(C) # 0}, and let S := {f : On + On : supp(f) finite}. For f,g E S define f 5 g : e (3 : O n + On)[ h one-to-one A (V() f(() 5 g ( h ( ( ) ) ] . A function $ : S -+ On is called monotonic increasing, if f(() 5 $(f) and if f 5 g implies $(f) 5 $(g). For a mapping $ : S -+ On let Clw(0) be the least set T of ordinals which contains 0 as an element and which is closed under the following rule: If f E S, range(f) c T and supp(f) C T, then $(f) E T. Let cp be the enumeration function of the class {( : (37)" = d]}. Let @ be the induced Sch~tte-KIanimertenrrfunction (see [Ill) which generates the Schiitte-Veblen-hierarchy of ordinals. Then CP is monotonic increasing. We show: If $ : S -+ On is monotonic increasing, then otyp(Cl+(O)) 5 min{( : ( = @(It)}, where lc(a) = 1 if Q = ( and lc(a) = 0 if a # (. For r an ordinal let S r := {f E S : supp(f) 2 r}. A function $ : S , 4 On is called r-monotonic increasing if f(() 5 $(f) and if f 5 g implies $(f) 5 $(g). For a function $ : Sr + O n let Cl+(O) 1 r be the least set T of ordinals, which contains 0 as an element, such that i f f E S, and range(f) c T, then $(f) E T. We show: Zf r 2 2 and if $ : S , -+ O n is r-monotonic increasing, then otyp(Cl+(O) I r) 5@ ( l r ) . We also prove a generalization of this theorem in terms of well-partial orderings. MSC: 03F15, 03E10, 06A06.

A Computation of the Maximal Order Type of the Term Ordering on Finite Multisets

“…Moreover since o(M (ε 0 ⊗ Ω)) = ω ε0⊗Ω we find the result o(T 1 ε0 ) = ϑ(Ω ε0 ) from [20]. Here the class T 1 ε0 from [20] corresponds to T (M (ε 0 ⊗ Ω)) in the current setting.…”

An order-theoretic characterization of the Howard–Bachmann-hierarchy

“…Definition 2. 26. Define the partial ordering T 2 as the subset of (T 2 , ≤ gap ) which consists of all finite rooted trees such that nodes with label 0 has zero or one immediate successor(s) and nodes with label 1 has exactly two immediate successors.…”

“…Weiermann's proof made essential use of the linearity of ordinals and did not generalize to partially ordered structures. Extending Schmidt's work in [26], the third author provided in a first step, an order-theoretic characterization for the large Veblen ordinal ϑΩ Ω . Quite recently, the authors of this paper were able to provide in [23] much more convincing methods and results which were suitable for being extended to larger ordinals as well.…”

Well-partial-orderings and the big Veblen number

“…This line of research has been taken up in [14] where the last author extended Schmidt's approach to transfinite arities. In more detail, motivated by order-theoretic properties of the functions considered by Veblen and Schütte (see, for example, [10,12] for further details), a well-partial-ordering (which we would denote in this article by T (M ⋄ (τ ×·))) has been considered, which corresponds to the ordinal ϑ (Ω τ ) using the ordinal notation system of [7].…”

“…In [14] it is shown that the maximal order type of T (M ⋄ (τ × ·)) is bounded by ϑ (Ω τ ) so that it can give rise to an ordinal notation system for ϑ (Ω τ ). Furthermore (by allowing the case τ = Ω ), it has been indicated in [14] that the order type T (M ⋄ (Ω × ·)) is bounded by the big Veblen number ϑ (Ω Ω ).…”