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（pca）的完备化，特别是克莱尼第二模型 $\mathcal{K}_2$ 及其推广形式。我们考虑先前研究过的嵌入性和完备化的弱概念与强概念。根据克洛普的一个结果，已知并非每个pca都具有强完备化。对 $\mathcal{K}_2$ 的完备化研究推论出弱嵌入与强嵌入是不同的，并且每个可数pca都具有弱完备化。接着，我们考虑更大基数下 $\mathcal{K}_2$ 的推广形式，并利用这些推广来证明：每个pca都具有弱完备化这一结论是一致的。