Well-partial-orderings and the big Veblen number

Abstract

In this article we characterize a countable ordinal known as the big Veblen number in terms of natural well-partially ordered tree-like structures. To this end, we consider generalized trees where the immediate subtrees are grouped in pairs with address-like objects. Motivated by natural ordering properties, extracted from the standard notations for the big Veblen number, we investigate different choices for embeddability relations on the generalized trees. We observe that for addresses using one finite sequence only, the embeddability coincides with the classical tree embeddability, but in this article we are interested in more general situations (transfinite addresses and wellpartially ordered addresses). We prove that the maximal order type of some of these new embeddability relations hit precisely the big Veblen ordinal ϑ Ω Ω . Somewhat surprisingly, changing a little bit the well-partially ordered addresses (going from multisets to finite sequences), the maximal order type hits an ordinal which exceeds the big Veblen number by far, namely ϑ Ω Ω Ω . Our results contribute to the research program (originally initiated by Diana Schmidt) on classifying properties of natural well-orderings in terms of order-theoretic properties of the functions generating the orderings.

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

“…This is done from 'below' and is treated in this section. The next theorems are generalizations of Theorems 9 and 10 in [23]. The proofs follow the same procedures as in that article, but they are more involved.…”

“…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. This recent approach is already far reaching but still misses essential ingredients for a order-theoretic characterization of η 0 .…”

“…In this subsection, we give the definitions of these wpo's. The reader can also find this kind of wpo's in [23] and [27]. For the actual proofs of well-partial-orderedness and the maximal order types, the reader has to wait until sections 3 and 4.…”

Ordinal notation systems corresponding to Friedman’s linearized well-partial-orders with gap-condition

“…It is our general belief that this is a maximal linear extension. In [18,19] we already obtained partial results concerning this conjecture. In this paper, we want to investigate whether this is also true for the linearized version of the gap-embeddability relation, i.e.…”

Statures and Sobrification Ranks of Noetherian Spaces

“…We compute the statures and dimensions of finite T 0 spaces, and of wellfounded chains in various topologies, in Section 6. We do the same for spaces with a cofinite topology in Section 7, for topological sums in Section 8, for lexicographic sums in Section 9, for topological products in Section 10, for Hoare powerspaces and powersets in Section 11, for spaces of finite words with the so-called word topology in Section 12 (generalizing the case of wpos of words explored by de Jongh and Parikh [33] and Schmidt [30]), for spaces of so-called heterogeneous words in the prefix topology in Section 13, and for spaces of finite multisets in Section 14 (generalizing the case of wpos of multisets explored by Aschenbrenner and Pong [2], Weiermann [35] and van der Meeren, Rathjen and Weiermann [34]).…”