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.

翻译：Ad\'amek 的古老理论构建了初始代数, 用于通过对古典立体理论的正方形交替迭代, 足够连续的异端体的初始代数。 我们证明一个新版本在建设性逻辑中有效, 使用“ 通缩” 迭代, 用于一个大小概念, 该概念摘自限序体, 仅仅是其中转性、 定向和有充分根据的属性。 我们从泰勒对正态的建设性处理中借款, 表明在任何特定指数的标志上界中存在大小。 从这一点可以推论, 新理论应用的内端异体有很多, 提供了一种薄弱的选择形式( 范登伯格、 摩尔季克、 帕尔姆格伦 和斯特赖赫 ), 例如, 以许多基本参数的内部逻辑 。