- April 2015
A general extension theorem for complete partial orders
An invocation of Zorn’s Lemma (ZL) often takes place within an indirect proof of a universal statement. Supposing towards a contradiction that there be any counterexample, the maximal or minimal counterexample provided by ZL helps–by a one-step argument–to the desired contradiction. Crucially though, this one step does not depend on maximality, and in fact a more general method is hovering in the background, which hitherto has been applied to hypothetical counterexamples only.
As a consequence, this and related proof patterns can sometimes be turned into direct proofs with Open Induction (OI) as an equivalent of ZL. We now bring this approach to a somewhat unexpected type of application: to extension theorems such as the ones going back to Helly, Hahn, Banach and Riesz as well as to Baer’s Criterion for whether a module is injective.
To this end, we distill a General Extension Theorem (GET) for complete partial orders, the intended meaning being that the poset under consideration consists of partial extensions of which one is to be proved total. The principal hypothesis of GET encodes the one-step argument from before: that there be a function extending each partial extension by any potential element of its domain. As compared with the typical indirect proof with ZL of an extension theorem, GET already postulates the existence of a total extension rather than a maximal one.
In ZF set theory, say, GET is an immediate consequence of OI, and both ZL and the Well-Ordering Theorem follow from GET in a straightforward way; whence GET is a ZF equivalent of the Axiom of Choice (AC). Attempting to deduce AC directly from GET has brought us to make an interesting move: the family of non-empty sets for which a choice function is sought needs to be replaced by a family of pointed sets! Needless to say, this has to be done extremely carefully, in order to ensure that AC does not follow trivially. This is joint work with Peter Schuster (Verona).