Numerals and ordinals occupy a special place in the typology of suppletion. In generative work, one basic cross-linguistic parameter is whether ordinal allomorphy displays internal vs. external marking. Internal marking is when irregular forms propagate from lower ordinals to higher ones (English ‘first’$ \to $‘twenty-first’), whereas external marking is the lack of propagation. We catalog ordinal formation in Armenian dialects through both formal-generative and functional-typological perspectives. We find that Eastern Armenian and Early Western Armenian are uniformly external-marking systems for the ordinals of ‘1–4’. However, Modern Western Armenian is a mixed system: ‘1’ displays external-marking while ‘2–4’ display internal-marking. Simultaneously, the ordinal of ‘1’ uses a suppletive portmanteau, while the ordinals of ‘2–4’ use agglutinative allomorphs. We formalize these differences in a derivational approach to morphology (Distributed Morphology). We argue that mixed systems arise from allomorphy rules that are sensitive to either constituency or linearity. The Western mixed system seems typologically rare and novel. Given our formal analysis, we then uncover other asymmetries in the propagation of irregular ordinals and the retention of portmanteau morphology across 35 Armenian varieties. The end result is a strong functional correlation between suppletion, external marking, and lower numerals.

We describe an organizing framework for the study of infinitary combinatorics. This framework is Čech cohomology. It describes ZFC principles distinguishing among the ordinals of the form $\omega _n$ . More precisely, this framework correlates each $\omega _n$ with an $(n+1)$ -dimensional generalization of Todorcevic’s walks technique, and begins to account for that technique’s “unreasonable effectiveness” on $\omega _1$ .

We show in contrast that on higher cardinals $\kappa $ , the existence of these principles is frequently independent of the ZFC axioms. Finally, we detail implications of these phenomena for the computation of strong homology groups and higher derived limits, deriving independence results in algebraic topology and homological algebra, respectively, in the process.

Abstract prepared by Jeffrey Bergfalk.

E-mail: jeffrey.bergfalk@univie.ac.at

The following bare-bones story introduces the idea of a cumulative hierarchy of pure sets: ‘Sets are arranged in stages. Every set is found at some stage. At any stage S: for any sets found before S, we find a set whose members are exactly those sets. We find nothing else at S’. Surprisingly, this story already guarantees that the sets are arranged in well-ordered levels, and suffices for quasi-categoricity. I show this by presenting Level Theory, a simplification of set theories due to Scott, Montague, Derrick, and Potter.

We prove that every automorphism of an infinite-dimensional vector space over a field is the product of four involutions, a result that is optimal in the general case. We also characterize the automorphisms that are the product of three involutions. More generally, we study decompositions of automorphisms into three or four factors with prescribed split annihilating polynomials of degree $2$.

Let λ be a regular ordinal with λ≥ω1. Then we prove that (λ+1)×λ is not base-countably metacompact. This implies that base-κ-paracompactness is not an inverse invariant of perfect mappings, which answers a question asked by Yamazaki.

We describe Wadge degrees of ω-languages recognizable by deterministic Turing machines. In particular, it is shown that the ordinal corresponding to these degrees is ξω where ξ = ω1CK is the first non-recursive ordinal known as the Church–Kleene ordinal. This answers a question raised in [2].

We apply set-theoretical ideas to an iteration problem of dynamical systems. Among other results, we prove that these iterations never stabilise later than the first uncountable ordinal; for every countable ordinal we give examples in Baire space and in Cantor space of an iteration that stabilises exactly at that ordinal; we give an example of an iteration with recursive data which stabilises exactly at the first non-recursive ordinal; and we find new examples of complete analytic sets simply definable from concepts of recurrence.

2000 Mathematics Subject Classification: primary 03E15, 37B20, 54H05; secondary 37B10, 37E15.