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 well-partially 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