Constructive order theory (Q2720327)

An upper bound \(s\) of a subset \(Y\) of a poset \(P\) is called a constructive supremum of \(Y\) if there exists a function \(\psi: P\to Y\) such that \(s\leq x\Leftrightarrow \psi(x)\leq x\) for all \(x\in P\). It is proved that the axiom of choice is equivalent to the postulate that every supremum is constructive, and also to the hypothesis that every directed set admits a function assigning to each finite subset an upper bound.NEWLINENEWLINENEWLINEA nonempty poset \(D\) is said to constructively directed if there exists a constructive direction for \(D\), that is, a function assigning to each nonempty finite subset of \(D\) an upper bound (in \(D\)). For this it is proved: The existence of a constructive direction for each directed poset is equivalent to the axiom of choice. Further, the set-theoretical induction principle (SIP) is investigated. It states that any system of sets that is closed under unions of well-ordered subsystems and contains all finite subsets of a given set must also contain that set itself. The axiom of multiple choice (MC) implies SIP. A lot of statements are listed which are all equivalent to SIP (and thus a consequence of MC). These concern inductivity, compactness, algebraic closure systems, and the exchange between chains and directed sets.