Constructing Initial Algebras Using Inflationary Iteration

An old theorem of Ad\'amek constructs initial algebras for sufficiently cocontinuous endofunctors via transfinite iteration over ordinals in classical set theory. We prove a new version that works in constructive logic, using "inflationary" iteration over a notion of size that abstracts from limit ordinals just their transitive, directed and well-founded properties. Borrowing from Taylor's constructive treatment of ordinals, we show that sizes exist with upper bounds for any given signature of indexes. From this it follows that there is a rich collection of endofunctors to which the new theorem applies, provided one admits a weak form of choice (the WISC axiom of van den Berg, Moerdijk, Palmgren and Streicher) that is known to hold, for example, in the internal logic of very many kinds of elementary topos.