  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 class of endofunctors to which the new theorem applies,
provided one admits a weak form of choice (WISC) due to Streicher, Moerdijk,
van den Berg and Palmgren, and which is known to hold in the internal
constructive logic of many kinds of topos.
