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An old theorem of Ad\'amek constructs initial algebras for sufficientlycocontinuous endofunctors via transfinite iteration over ordinals in classicalset theory. We prove a new version that works in constructive logic, using"inflationary" iteration over a notion of size that abstracts from limitordinals just their transitive, directed and well-founded properties. Borrowingfrom Taylor's constructive treatment of ordinals, we show that sizes exist withupper bounds for any given signature of indexes. From this it follows thatthere 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 internalconstructive logic of many kinds of topos.

Constructing Initial Algebras Using Inflationary Iteration