Halin in 1965 proved that if a graph has $n$ many pairwise disjoint rays for each $n$ then it has infinitely many pairwise disjoint rays. We analyze the complexity of this and other similar results in terms of computable and proof theoretic complexity. The statement of Halin’s theorem and the construction proving it seem very much like standard versions of compactness arguments such as König’s Lemma. Those results, while not computable, are relatively simple. They only use arithmetic procedures or, equivalently, finitely many iterations of the Turing jump. We show that several Halin-type theorems are much more complicated. They are among the theorems of hyperarithmetic analysis. Such theorems imply the ability to iterate the Turing jump along any computable well ordering. Several important logical principles in this class have been extensively studied beginning with work of Kreisel, H. Friedman, Steel and others in the 1960s and 1970s. Until now, only one purely mathematical example was known. Our work provides many more and so answers Question 30 of Montalbán’s Open Questions in Reverse Mathematics in 2011. Some of these theorems including ones in Halin in 1965 are also shown to have unusual proof theoretic strength as well.

Halin 在 1965 年证明，如果一个图有$n$每个都有许多成对不相交的射线$n$那么它有无穷多个成对不相交的射线。我们根据可计算和证明理论的复杂性来分析这个结果和其他类似结果的复杂性。哈林定理的陈述和证明它的构造看起来非常像柯尼希引理等紧性论证的标准版本。这些结果虽然不可计算，但相对简单。他们只使用算术过程，或者等效地，图灵跳跃的有限多次迭代。我们证明了几个 Halin 型定理要复杂得多。它们属于超算术分析的定理。这些定理意味着能够沿着任何可计算的井序迭代图灵跳跃。从 Kreisel、H. Friedman、Steel 等人在 20 世纪 60 年代和 1970 年代的工作开始，本课程中的几个重要逻辑原理已得到广泛研究。到目前为止，人们只知道一个纯数学的例子。我们的工作为 2011 年 Montalbán 的逆向数学开放问题第 30 题提供了更多答案。其中一些定理（包括 1965 年哈林的定理）也被证明具有不同寻常的证明理论强度。