A New Construction Principle
Tapani Hyttinen, Gianluca Paolini, Davide Emilio Quadrellaro
TL;DR
This work develops a general Construction Principle CP(K,*) within the setting of weak Abstract Elementary Classes to capture a broad non-smoothness phenomenon that obstructs infinitary axiomatisability. Building on canonical amalgamation, universality, and κ^+-categoricity, the authors derive CP(K,*) and CP^κ_{λ,δ}(K,≤) as tools to construct $\mathfrak{L}_{\infty,\kappa^+}$-free structures outside a given class, prove their equivalence to non-axiomatizability in $\mathfrak{L}_{\infty,\omega_1}$ in ZFC, and, under $V=L$, non-axiomatizability in $\mathfrak{L}_{\infty,\infty}$. They instantiate CP(K,*) in concrete contexts beyond varieties of algebras, including free products of cyclic groups, direct sums of rank-1 torsion-free abelian groups, and incidence-geometric structures (Steiner systems and generalized polygons), thereby obtaining new non-axiomatizability results and uncountable categoricity phenomena for free incidence structures. The paper unifies and extends Eklof–Mekler–Shelah’s Construction Principle to a broad abstract framework, showing that failure of Smoothness and canonical amalgamation patterns yield strong model-theoretic nondefinability results with wide applicability. Overall, the work broadens the landscape where infinitary logic and AECs interact with concrete combinatorial and geometric constructions, highlighting the deep connections between categoricity, amalgamation, and axiomatisability.
Abstract
We use the framework of Abstract Elementary Classes ($\mathrm{AEC}$s) to introduce a new Construction Principle $\mathrm{CP}(\mathbf{K},\ast)$, which strictly generalises the Construction Principle of Eklof, Mekler and Shelah and allows for many novel applications beyond the setting of universal algebra. In particular, we show that $\mathrm{CP}(\mathbf{K},\ast)$ holds in the classes of free products of cyclic groups of fixed order, direct sums of a fixed torsion-free abelian group of rank 1 which is not $\mathbb{Q}$, (infinite) free $(k,n)$-Steiner systems, and (infinite) free generalised $n$-gons. From this we derive, in ZFC, that these classes of structures are not axiomatisable in the logic $\mathfrak{L}_{\infty,ω_1}$, and, under $V=L$, that they are not axiomatisable in $\mathfrak{L}_{\infty,\infty}$.