The notion of pure extension of modules, and the corresponding notions of pure-injectivity and pure-projectivity, play a central role in commutative algebra, aspecially in its connections to model theory. In joint work with Casarosa, motivated by applications to algebraic topology and operator algebras, we have introduced the first “higher order” generalization of purity, defining “pure extensions of order α” for every countable ordinal α. Classical purity corresponds to the particular case when α=0.

In this talk I will present connections with logic: as classical purity is characterized in terms of first order logic, we will show that higher order purity has a natural interpretation in terms of infinitary logic, in such a way that the order of purity corresponds to length of infinitary formulae.