The Aleph Zero or Zero Dichotomy

The Aleph Zero or Zero Dichotomy is a strong version of Zeno’s Dichotomy II which being entirely derived from the topological successiveness of the ω∗-order comes to the same Zeno’s absurdity. 1. Zeno’s paradoxes and modern science Zeno’s Paradoxes have interested philosophers of all times (see [13], [14], [79], [68], [47] or [24] for historical background), although until the middle of the XIX century they were frequently considered as mere sophisms [13], [14], [67], [68]. From that time, and particularly along the XX century, they became the unending source of new philosophical, mathematical and physical discussions. Authors as Hegel [43], James [48], Russell [67], Whitehead [81], [82] or Bergson [9], [10] focused their attention on the challenging world of Zeno’s paradoxes. At the beginning of the second half of the XX century the pioneering works of Black [11], Wisdom [83], Thomson [75], [76], and Benacerraf [8] introduced a new way of discussing the possibilities to perform an actual infinity of actions in a finite time (a performance which is involved in most of Zeno’s paradoxes). I refer to Supertask Theory [64]. In fact, infinity machines, or supermachines, are our modern Achilles substitutes. A supermachine is a theoretical device supposedly capable of performing countably many actions in a finite interval of time. The possibilities of performing an uncountable infinity of actions were ruled out by P. Clark and S. Read [22], for which they made use of a Cantor’s argument on the impossibility of dividing a real interval into uncountably many adjacent parts [19]. Although supertasks have also been examined from the perspective of nonstandard analysis ([55], [54], [1], [52]), as far as I know the possibilities to perform an hypertask along an hyperreal interval of time have not been discussed, although finite hyperreal intervals can be divided into uncountable many successive infinitesimal intervals, the so called hyperfinite partitions ([73], [34], [49], [44], etc.). Supertask theory has finally turned its attention, particularly from the last decade of the XX century, towards the discussion of the physical plausibility of supertasks ([66], [60], [64], [68], [39], [41], [40]) as well as on the implications of supertasks in the physical world ([60], [61], [62], [30], [63], [58], [2], [3], [65]), including relativistic and quantum mechanics perspectives [80], [45], [28], [29], [58], [27], [70] During the last half of the XX century several solutions to some of Zeno’s paradoxes have been proposed. Most of these solutions were found in the context of new branches of mathematics as Cantor’s transfinite arithmetic, topology, measure theory [37], [38], [85], [39], [41], [40], and more recently internal set theory (a branch of nonstandard analysis) [55], [54]. It is also worth noting the solutions proposed by P. Lynds within a classical and quantum mechanics framework [51]. Some of these solutions, however, have been contested [59], [1]. And in most of cases the proposed solutions do not explain where Zeno’s arguments fail [59], [64]. Moreover, some of the proposed solutions gave rise to a significant collection of new and exciting problems [68], [47] [70].