Ekeland’s variational principle in weak and strong systems of arithmetic

We analyze Ekeland’s variational principle in the context of reverse mathematics. We find that that the full variational principle is equivalent to $$\Pi ^1_1\text{- }\mathsf {CA}_0$$ Π 1 1 - CA 0 , a strong theory of second-order arithmetic, while natural restrictions (e.g. to compact spaces or to continuous functions) yield statements equivalent to weak König’s lemma ($$\mathsf {WKL}_0$$ WKL 0 ) and to arithmetical comprehension ($$\mathsf {ACA}_0$$ ACA 0 ). We also find that the localized version of Ekeland’s variational principle is equivalent to $$\Pi ^1_1\text{- }\mathsf {CA}_0$$ Π 1 1 - CA 0 , even when restricted to continuous functions. This is a rare example of a statement about continuous functions having great logical strength.

Constructing sequences one step at a time

We propose a new method for constructing Turing ideals satisfying principles of reverse mathematics below the Chain–Antichain ([Formula: see text]) Principle. Using this method, we are able to prove several new separations in the presence of Weak König’s Lemma ([Formula: see text]), including showing that [Formula: see text] does not imply the thin set theorem for pairs, and that the principle “the product of well-quasi-orders is a well-quasi-order” is strictly between [Formula: see text] and the Ascending/Descending Sequences principle, even in the presence of [Formula: see text].

The Brouwer invariance theorems in reverse mathematics

In [12], John Stillwell wrote, ‘finding the exact strength of the Brouwer invariance theorems seems to me one of the most interesting open problems in reverse mathematics.’ In this article, we solve Stillwell’s problem by showing that (some forms of) the Brouwer invariance theorems are equivalent to the weak König’s lemma over the base system ${\sf RCA}_0$ . In particular, there exists an explicit algorithm which, whenever the weak König’s lemma is false, constructs a topological embedding of $\mathbb {R}^4$ into $\mathbb {R}^3$ .

Connected choice and the Brouwer fixed point theorem

We study the computational content of the Brouwer Fixed Point Theorem in the Weihrauch lattice. Connected choice is the operation that finds a point in a non-empty connected closed set given by negative information. One of our main results is that for any fixed dimension the Brouwer Fixed Point Theorem of that dimension is computably equivalent to connected choice of the Euclidean unit cube of the same dimension. Another main result is that connected choice is complete for dimension greater than or equal to two in the sense that it is computably equivalent to Weak Kőnig’s Lemma. While we can present two independent proofs for dimension three and upward that are either based on a simple geometric construction or a combinatorial argument, the proof for dimension two is based on a more involved inverse limit construction. The connected choice operation in dimension one is known to be equivalent to the Intermediate Value Theorem; we prove that this problem is not idempotent in contrast to the case of dimension two and upward. We also prove that Lipschitz continuity with Lipschitz constants strictly larger than one does not simplify finding fixed points. Finally, we prove that finding a connectedness component of a closed subset of the Euclidean unit cube of any dimension greater than or equal to one is equivalent to Weak Kőnig’s Lemma. In order to describe these results, we introduce a representation of closed subsets of the unit cube by trees of rational complexes.

COMPUTABILITY IN UNCOUNTABLE BINARY TREES

Computability, while usually performed within the context of ω, may be extended to larger ordinals by means of α-recursion. In this article, we concentrate on the particular case of ω1-recursion, and study the differences in the behavior of ${\rm{\Pi }}_1^0$-classes between this case and the standard one.Of particular interest are the ${\rm{\Pi }}_1^0$-classes corresponding to computable trees of countable width. Classically, it is well-known that the analog to König’s Lemma—“every tree of countable width and uncountable height has an uncountable branch”—fails; we demonstrate that not only does it fail effectively, but also that the failure is as drastic as possible. This is proven by showing that the ω1-Turing degrees of even isolated paths in computable trees of countable width are cofinal in the ${\rm{\Delta }}_1^1\,{\omega _1}$-Turing degrees.Finally, we consider questions of nonisolated paths, and demonstrate that the degrees realizable as isolated paths and the degrees realizable as nonisolated ones are very distinct; in particular, we show that there exists a computable tree of countable width so that every branch can only be ω1-Turing equivalent to branches of trees with ${\aleph _2}$-many branches.

ON THE UNIFORM COMPUTATIONAL CONTENT OF RAMSEY’S THEOREM

We study the uniform computational content of Ramsey’s theorem in the Weihrauch lattice. Our central results provide information on how Ramsey’s theorem behaves under product, parallelization, and jumps. From these results we can derive a number of important properties of Ramsey’s theorem. For one, the parallelization of Ramsey’s theorem for cardinalityn≥ 1 and an arbitrary finite number of colorsk≥ 2 is equivalent to then-th jump of weak Kőnig’s lemma. In particular, Ramsey’s theorem for cardinalityn≥ 1 is${\bf{\Sigma }}_{n + 2}^0$-measurable in the effective Borel hierarchy, but not${\bf{\Sigma }}_{n + 1}^0$-measurable. Secondly, we obtain interesting lower bounds, for instance then-th jump of weak Kőnig’s lemma is Weihrauch reducible to (the stable version of) Ramsey’s theorem of cardinalityn+ 2 forn≥ 2. We prove that with strictly increasing numbers of colors Ramsey’s theorem forms a strictly increasing chain in the Weihrauch lattice. Our study of jumps also shows that certain uniform variants of Ramsey’s theorem that are indistinguishable from a nonuniform perspective play an important role. For instance, the colored version of Ramsey’s theorem explicitly includes the color of the homogeneous set as output information, and the jump of this problem (but not the uncolored variant) is equivalent to the stable version of Ramsey’s theorem of the next greater cardinality. Finally, we briefly discuss the particular case of Ramsey’s theorem for pairs, and we provide some new separation techniques for problems that involve jumps in this context. In particular, we study uniform results regarding the relation of boundedness and induction problems to Ramsey’s theorem, and we show that there are some significant differences with the nonuniform situation in reverse mathematics.