We investigate completions of partial combinatory algebras (pcas), in particular of Kleene's second model $\mathcal{K}_2$ and generalizations thereof. We consider weak and strong notions of embeddability and completion that have been studied before. By a result of Klop it is known that not every pca has a strong completion. The study of completions of $\mathcal{K}_2$ has as corollaries that weak and strong embeddings are different, and that every countable pca has a weak completion. We then consider generalizations of $\mathcal{K}_2$ for larger cardinals, and use these to show that it is consistent that every pca has a weak completion.

翻译：我们研究部分组合代数（partial combinatory algebras, pcas）的完备化，特别是克莱尼第二模型$\mathcal{K}_2$及其推广。我们考虑了此前研究过的嵌入与完备化的弱概念与强概念。根据Klop的一个结果，并非每个pca都有强完备化。对$\mathcal{K}_2$完备化的研究推论出弱嵌入与强嵌入是不同的，并且每个可数pca都有弱完备化。进而，我们考虑$\mathcal{K}_2$对更大基数的推广，并利用这些推广证明：每个pca都有弱完备化是一致的。