Abstract:We study the pigeonhole principle for $\Sigma _2$-definable injections with domain twice as large as the codomain, and the weak König lemma for $\Delta ^0_2$-definable trees in which every level has at least half of the possible nodes. We show that the latter implies the existence of $2$-random reals, and is conservative over the former. We also show that the former is strictly weaker than the usual pigeonhole principle for $\Sigma _2$-definable injections.

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中文翻译：

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鸽巢原理与 König 引理相遇的地方

摘要：我们研究了 $\Sigma _2$ 可定义注入的鸽巢原理，域是 codomain 的两倍，以及 $\Delta ^0_2$ 可定义树的弱 König 引理，其中每个级别至少有一半可能的节点。我们证明后者意味着 $2$-随机实数的存在，并且比前者保守。我们还表明，对于 $\Sigma _2$ 可定义的注射，前者比通常的鸽巢原理要弱得多。

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