Barrington Bayley SF Gateway Omnibus: The Soul of the Robot, The Knights of the Limits, The Fall of Chronopolis

At first his meaning escaped me. Number was another way of classifying the innumerable kinds of space in the universe, he explained. There was at least one space for every possible number (a theorem stated that there were more spaces than one for every possible number), and they were arranged in an ascending series, each space having its ‘centre of gravity’ about a particular number. We are near to the bottom of the scale as our ‘centre of gravity’ is the number One (there are spaces preferring fractions and at least one preferring Zero). The consequences are immediate and self-evident: singleness is what signifies a complete object in our world; integral unity is all, and the state of there being two of a thing is incidental – a thing comes into its own when it is one. We all accept this innately. Every entity and thing is itself by assigning the number One to itself. Higher numbers introduce additional qualities, but do not carry the same weight as one.^*

In the space next above us in the scale completeness attaches to the number two. ‘Two-ness’ is ideal, and singleness is incomplete in the same way that a fraction or a part is incomplete in our world. I reflected on what a mass migration there would be if communication could be established with that world – for we also have the yearning after two in our shadowy, tantalised way. Our lives are full of complementary pairs. The tragedy of lovers is that they are thwarted by the One-ness of the spatial system: each remains alone and solitary, however much they strive and strain to be completely merged as two – for the vain yearning of lovers is not to be made One, which would negate the whole proceeding, but to be, as it were, indistinguishably blended as Two. Should a pair of Eros-struck lovers by some magic or science transpose themselves to this other realm where Two is All, then their bliss would be beyond describing.

More remotely, other worlds model themselves on Three, Four, Five, and so on up the scale of integers to infinity. In addition there is a corresponding scale of negative integers, as also of worlds modelled on every possible fraction, on irrational numbers, on imaginary numbers, and on groups, sets and series of numbers, such as on all the primes, all the odd integers, all the even integers, and on arithmetic and geometric progressions. Beyond even these abstruse factualities are the ranges of worlds, centred on numbers and number systems not possible or conceivable to us. The only truly symmetrical, non-centred, relativistic space-time, said the Knight, is one giving equal weight to all numbers.

Georg Cantor, wrestling with the enigma of the infinite, discovered a branch of mathematics called transfinite arithmetic, in which he developed a progression of numbers analogous to the positive integers but whose first term was infinity and whose succeeding terms were as qualitatively different from and beyond infinity as are Two, Three, Four, etc., beyond One. In short, he found that there are numbers larger than infinity. As might be expected, the Knight confirmed the reality of this number system and of the transfinite space that goes with it. There is a whole range of transfinite spaces, probably even larger than the range of finite and infinite spaces (since the number of total spaces is both finite, infinite and transfinite). At this point the Knight seemed to think that we were wandering from the type of description from which I might be expected to profit, and proposed to resume, expatiating on those nearer to familiarity. I objected; it was diverting, but less challenging, to be presented with nothing but modifications of an existence I already knew. In a sense I could almost have invented these modifications myself. Would not the Knight consent to offer me, or at least attempt to make me understand, worlds having no common ground with my own – for even the Knight’s own locational-transitional space-time, I reminded him, was not hard to describe. I longed to hear something so original as to blow my mind free of all its preconceptions. After some hesitation and muttering as to the perplexities engendered by my request, the Knight agreed to make the effort and favoured me with the following amazing descriptions:^*

Suddenly the Knight broke off to warn me that the power drain was now significantly close to tolerable limits and that he would not be able to linger much longer. A brief feeling of panic assailed me. There must be one question that above all others needed to be asked – yes! The choice was obvious, and I did not delay in putting it. Did the chess-people have any single, particular purpose in undertaking their admirable explorations of space?

The instinct of exploration, said the Knight, is a natural one. There was a central quest, however: to try to determine whether, in the multiplicity of space-times, there is a common universal law or principle, and thereby to discover how existence originates and is maintained.