The Bolzano-Weierstrass Theorem is the Jump of Weak König's Lemma

Abstract

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