A Disturbing Supertask

This paper examines the consistency of ω-order by means of a supertask that functions as a supertrap for the assumed existence of ω-ordered collections, which are simultaneously complete (as is required by the Actual infinity) and uncompletable (because no last element completes them). As Cantor himself proved [2], [3], ω-order is a formal consequence of assuming the existence of denumerable sets as complete totalities. Although it is hardly recognized, to be ω-ordered means to be both complete and uncompletable. In fact, the Axiom of Infinity states the existence of complete denumerable totalities, the most simple of which are ω-ordered, i.e. with a first element and such that each element has an immediate successor. Consequently, there is not a last element that completes ω-ordered totalities. To be complete and uncompletable may seem a modest eccentricity in the highly eccentric infinite paradise of our days, but its simplicity is just an advantage if we are interested in examining the formal consistency of ω-order. In addition, ω is the first transfinite ordinal, the one on which all successive transfinite ordinals are built up. This magnifies the interest of examining its formal consistency. The short discussion that follows is based on a supertask conceived to put into question just the ability of being complete and uncompletable that characterizes ω-order. 1. The last disk: a disturbing supertask Consider a hollow cylinder C and an ω-ordered collection of identical disks 〈di〉i∈N such that each disk di fits exactly within the cylinder. For the sake of clarity, we will assume that a disk d0 is initially placed inside the cylinder, although this is irrelevant to our discussion. Let ai be the action of replacing disk di−1 inside the cylinder by its immediate successor disk di, which is accomplished by placing di completely within the cylinder. Consider the ω-ordered sequence of actions 〈ai〉i∈N and assume that each action ai is carried out at instant ti, being ti an element of the ω-ordered sequence of instants 〈ti〉i∈N in the real interval [ta, tb) such that: