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 $\mathfrak{ L}_{ \infty,\omega_1}$, and, under $V=L$, that they are not axiomatisable in $\mathfrak{ L}_{ \infty,\infty}$.
Categories: math.LO
