Halin's infinite ray theorems: Complexity and reverse mathematics

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 Konig'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 Montalban'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.

Automated summary

This paper examines the complexity of Halin's theorem and similar results in terms of computable and proof theoretic complexity. It finds that these theorems are more complicated than standard compactness arguments and belong to the theorems of hyperarithmetic analysis, which imply the ability to iterate the Turing jump along any computable well ordering. Several important logical principles in this class have been extensively studied and only one purely mathematical example was previously known. The work presented in this paper provides many more examples and answers an open question in reverse mathematics.