Halin in 1965 proved that if a graph has [Formula: see text] many pairwise disjoint rays for each [Formula: see text] 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. (shrink)

We establish the decidability of the${{\rm{\Sigma }}_2}$theory of both the arithmetic and hyperarithmetic degrees in the language of uppersemilattices, i.e., the language with ≤, 0, and$\sqcup$. This is achieved by using Kumabe-Slaman forcing, along with other known results, to show given finite uppersemilattices${\cal M}$and${\cal N}$, where${\cal M}$is a subuppersemilattice of${\cal N}$, that every embedding of${\cal M}$into either degree structure extends to one of${\cal N}$iff${\cal N}$is an end-extension of${\cal M}$.

Theorems of hyperarithmetic analysis occupy an unusual neighborhood in the realms of reverse mathematics and recursion-theoretic complexity. They lie above all the fixed iterations of the Turing jump but below ATR $_{0}$. There is a long history of proof-theoretic principles which are THAs. Until the papers reported on in this communication, there was only one mathematical example. Barnes, Goh, and Shore [1] analyze an array of ubiquity theorems in graph theory descended from Halin’s [9] work on rays in graphs. They (...) seem to be typical applications of ACA $_{0}$ but are actually THAs. These results answer Question 30 of Montalbán’s Open Questions in Reverse Mathematics [19] and supply several other natural principles of different and unusual levels of complexity.This work led in [25] to a new neighborhood of the reverse mathematical zoo: almost theorems of hyperarithmetic analysis. When combined with ACA $_{0}$ they are THAs but on their own are very weak. Denizens both mathematical and logical are provided. Generalizations of several conservativity classes are defined and these ATHAs as well as many other principles are shown to be conservative over RCA $_{0}$ in all these senses and weak in other recursion-theoretic ways as well. These results answer a question raised by Hirschfeldt and reported in [19] by providing a long list of pairs of principles one of which is very weak over RCA $_{0}$ but over ACA $_{0}$ is equivalent to the other which may be strong or very strong going up a standard hierarchy and at the end being stronger than full second-order arithmetic. (shrink)