We classify the computational content of the Bolzano-Weierstraß Theorem and variants thereof in the Weihrauch lattice. For this purpose we first introduce the concept of a derivative or jump in this lattice and we show that it has some properties similar to the Turing jump. Using this concept we prove that the derivative of closed choice of a computable metric space is the cluster point problem of that space. By specialization to sequences with a relatively compact range we obtain a characterization of the Bolzano-Weierstraß Theorem as the derivative of compact choice. In particular, this shows that the Bolzano-Weierstraß Theorem on real numbers is the jump of Weak Kőnig's Lemma. Likewise, the Bolzano-Weierstraß Theorem on the binary space is the jump of the lesser limited principle of omniscience LLPO and the Bolzano-Weierstraß Theorem on natural numbers can be characterized as the jump of the idempotent closure of LLPO (which is the jump of the finite parallelization of LLPO). We also introduce the compositional product of two Weihrauch degrees f and g as the supremum of the composition of any two functions below f and g, respectively. Using this concept we can express the main result such that the Bolzano-Weierstraß Theorem is the compositional product of Weak Kőnig's Lemma and the Monotone Convergence Theorem. We also study the class of weakly limit computable functions, which are functions that can be obtained by composition of weakly computable functions with limit computable functions. We prove that the Bolzano-Weierstraß Theorem on real numbers is complete for this class. Likewise, the unique cluster point problem on real numbers is complete for the class of functions that are limit computable with finitely many mind changes. We also prove that the Bolzano-Weierstraß Theorem on real numbers and, more generally, the unbounded cluster point problem on real numbers is uniformly low limit computable. Finally, we also provide some separation techniques that allow to prove non-reducibilities between certain variants of the Bolzano-Weierstraß Theorem.

We determine the computational complexity of the Hahn-Banach Extension Theorem. To do so, we investigate some basic connections between reverse mathematics and computable analysis. In particular, we use Weak König's Lemma within the framework of computable analysis to classify incomputable functions of low complexity. By defining the multi-valued function Sep and a natural notion of reducibility for multi-valued functions, we obtain a computational counterpart of the subsystem of second order arithmetic WKL 0 . We study analogies and differences between WKL 0 and the class of Sep-computable multi-valued functions. Extending work of Brattka, we show that a natural multi-valued function associated with the Hahn-Banach Extension Theorem is Sep-complete.

We determine the computational complexity of the Hahn-Banach Extension Theorem. To do so, we investigate some basic connections between reverse mathematics and computable analysis. In particular, we use Weak König's Lemma within the framework of computable analysis to classify incomputable functions of low complexity. By defining the multivalued function Sep and a natural notion of reducibility for multivalued functions, we obtain a computational counterpart of the subsystem of second-order arithmetic WKL 0 . We study analogies and differences between WKL 0 and the class of Sep-computable multivalued functions. Extending work of Brattka, we show that a natural multivalued function associated with the Hahn-Banach Extension Theorem is Sep-complete.

We study the computability-theoretic complexity and proof-theoretic strength of the following statements: (1) "If X is a well-ordering, then so is ε X ", and (2) "If X is a well-ordering, then so is ϕ(α, X )", where α is a fixed computable ordinal and ϕ represents the two-placed Veblen function. For the former statement, we show that ω iterations of the Turing jump are necessary in the proof and that the statement is equivalent to ACA + 0 over RCA 0 . To prove the latter statement we need to use ω α iterations of the Turing jump, and we show that the statement is equivalent to Π 0 ω α -CA 0 . Our proofs are purely computability-theoretic. We also give a new proof of a result of Friedman: the statement "if X is a well-ordering, then so is ϕ(X , 0)" is equivalent to ATR 0 over RCA 0 ., Z(12), · · · Proof. Suppose first that Y = J (Z). When n = 0 let σ n = Z ↾ 1, which works by (P1). When n > 0 let. This is the base step of an induction that, using the same argument, shows that Y (i) = J(σ n )(i) for every i < n. Thus Y ↾ n ⊆ J(σ n ). By (P6), we have Y ↾ n ∈ J(N n. Now assume that (2) holds, and suppose towards a contradiction that Y = J (Z). Let n be least such that Y (n − 1) = J (Z)(n − 1). If σ n ⊂ Z is such that Y ↾ n = J(σ n ) we have J(σ n )(n−1) = J (Z)(n−1). This can occur only if {n−1} σn (n−1)↑ and {n − 1} Z (n − 1)↓, which implies n ′ > |σ n |, where n ′ = |J (Z)(n − 1)|. Notice that for any m > n ′ we have J(Z ↾ m)(n − 1) = J (Z)(n − 1) and hence J(Z ↾ m)(n − 1) = Y (n − 1). This contradicts the existence of σ n ′ ⊂ Z with |σ n ′ | > n ′ such that Y ↾ n ′ = J(σ n ′ ).The following corollary is obtained by iterating the Lemma.Corollary 4.6. For every m > 0, given Y, Z ∈ N N , the following are equivalent:(1) Y = J m (Z);(2) for every n there exists σ n ⊂ Z with |σ n | ≥ n+m such that Y ↾ n = J m (σ n ).The Jump function leads to the definition of the Jump Tree.Definition 4.7. Given a tree T ⊆ N n, and therefore the union above does actually produce Z. We have just proved the fol...

There are close similarities between the Weihrauch lattice and the zoo of axiom systems in reverse mathematics. Following these similarities has often allowed researchers to translate results from one setting to the other. However, amongst the big five axiom systems from reverse mathematics, so far has no identified counterpart in the Weihrauch degrees. We explore and evaluate several candidates, and conclude that the situation is complicated.