The first omega alephs: from simplices to trees of trees to higher walks

TL;DR

This paper provides new proofs and clarifications of the cohomological dimensions of certain ordinals, introducing higher-dimensional combinatorial structures like simplices, trees, and walks within ZFC set theory.

Contribution

It introduces novel higher-dimensional combinatorial characterizations of ordinals and generalizes existing techniques such as Todorcevic's walks within a ZFC framework.

Findings

New proof and mild strengthening of Mitchell's theorem on cohomological dimension.

Descriptions of higher-dimensional structures like simplices and trees for ordinals.

Development of a simple, finitely iterable technique for higher-dimensional combinatorics.

Abstract

The point of departure for the present work is Barry Mitchell's 1972 theorem that the cohomological dimension of $\aleph_n$ is $n+1$. We record a new proof and mild strengthening of this theorem; our more fundamental aim, though, is some clarification of the higher-dimensional infinitary combinatorics lying at its core. In the course of this work, we describe simplicial characterizations of the ordinals $\omega_n$, higher-dimensional generalizations of coherent Aronszajn trees, bases for critical inverse systems over large index sets, nontrivial $n$-coherent families of functions, and higher-dimensional generalizations of portions of Todorcevic's walks technique. These constructions and arguments are undertaken entirely within a $\mathsf{ZFC}$ framework; at their heart is a simple, finitely iterable technique of compounding $C$-sequences.