A New Construction Principle

TL;DR

This paper introduces a new construction principle within Abstract Elementary Classes that demonstrates certain uncountably categorical classes are not axiomatisable in specific infinitary logics, with applications to various algebraic and combinatorial structures.

Contribution

It generalizes existing construction principles using AECs, enabling new applications in model theory and showing non-axiomatisability results in ZFC and under $V=L$.

Findings

Several uncountably categorical classes are not axiomatisable in $rak{L}_{ ext{infty}, ext{omega}_1}$ in ZFC.

Under $V=L$, these classes are also not axiomatisable in $rak{L}_{ ext{infty}, ext{infty}}$.

Applications include free products of cyclic groups, direct sums of torsion-free abelian groups, Steiner systems, and generalized polygons.

Abstract

We use the framework of Abstract Elementary Classes ($\mathrm{AEC}$s) to introduce a new Construction Principle $\mathrm{CP}(\mathbf{K},\ast)$, which generalises the Construction Principle of Eklof, Mekler and Shelah and allows for many novel applications beyond the setting of universal algebra. From this we derive, in ZFC, that several uncountably categorical classes of structures are not axiomatisable in the logic $\mathfrak{L}_{\infty,\omega_1}$, and, under $V=L$, that they are not axiomatisable in $\mathfrak{L}_{\infty,\infty}$. In particular, our methods apply to: free products of cyclic groups of fixed order, direct sums of a fixed torsion-free abelian group of rank $1$ which is not $\mathbb{Q}$, free $(k,n)$-Steiner systems, and free generalised $n$-gons.