Peter Schuster
Reports on Mathematical Logic, Number 53, 2018, pp. 79-96
https://doi.org/10.4467/20842589RM.18.005.8838The typical indirect proof of an abstract extension theorem, by the Kuratowski-Zorn lemma, is based on a onestep extension argument. While Bell has observed this in case of the axiom of choice, for subfunctions of a given relation, we now consider such extension patterns on arbitrary directed-complete partial orders. By postulating the existence of so-called total elements rather than maximal ones, we can single out an immediate consequence of the Kuratowski-Zorn lemma from which quite a few abstract extension theorems can be deduced more directly, apart from certain definitions by cases. Applications include Baer’s criterion for a module to be injective. Last but not least, our general extension theorem is equivalent to a suitable form of the Kuratowski-Zorn lemma over constructive set theory.
Peter Schuster
Reports on Mathematical Logic, Number 47, 2012, pp. 63-86
https://doi.org/10.4467/20842589RM.12.003.0684The standard omniscience principles are interpreted in a systematic way within the context of binary trees. With this dictionary at hand we revisit the weak Konig lemma (WKL) and Brouwer’s fan theorem (FAN). We first study how one can arrive from FAN at WKL, and then give a direct decomposition, without coding, of WKL into the lesser limited principle of omniscience and an instance of the principle of dependent choices. As a complement we provide, among other equivalents of the standard omniscience principles, a uniform method to formulate most of them.