ASPECTS OF THE PROBLEM OF THE SPATIOTEMPORAL INFINITY OF THE WORLD

The problem of the spatiotemporal infinity of the Universe has a separate status among different problems of the infinity of the world addressed by physics, cosmology and philosophy. The problem itself can be subdivided into eight distinct, independently formulable problems. The first is the problem of the temporal infinity of the Universe, which can be stated as the question: is time infinite? Assuming a definite unit of time (for instance, a second or a year), we can formulate the problem as follows: is time composed of a finite (i.e. expressable with the aid of some natural number) or infinite (i.e. non-expressable in this way) number of the units of time (e.g. seconds)? The second problem is one of the genetic infinity of the Universe, which can be expressed as the question: does the world last a finite or an infinite number of time units? The problem of the spatial infinity of the Universe (is space infinite) is the third one. Having assumed a definite unit of space (e.g. cm3 or km3), one can ask a question: does space include a finite or an infinite number of such units (e.g. km3)? The fourth problem is the one of the material infinity of the Universe. It can be expressed in a twofold way: (a) does matter occupy a finite or an infinite space? Having assumed a definite unit for matter (any unit of mass such as gram or kilogram), one can formulate the problem as follows: (b) is matter composed of a finite or an infinite amount of such units (e.g. grams)? The question arises whether the two above-mentioned formulae of the fourth problem (a and b) are equivalent. It turns out that they are equivalent if once accept a natural assumption that is usual among the physicists of a density of matter being finite in any area. Should this assumption prove wrong, then, for instance, an infinite amount of matter would fit into a finite area, which would mean that the Universe is infinite in the sense (b) but finite in the sense (a). The problems identified above are the problems of the “extensional” infinity of the Universe. They appear when we pass from the macroscopic objects (or areas) to increasingly large ones, and ask if there is an “upper” limit for the existence of the objects. These problems deserve the label of the problems of cosmological infinity of the Universe. Directing our attention in the “opposite” direction, towards increasingly small objects, we arrive at four analogous problems of the infinity of the Universe “inwards”: the problems of the infinite divisibility of time, space and 354 Part Five: Philosophy of Physics and Cosmology matter (in temporal and spatial aspects). We shall call them the problems of the microphysical (or microcosmical) infinity of the Universe. The first of them (and the fifth in general series) is the problem of the infinite divisibility of time. Applying mathematical terminology, we can ask: is time a continuous set, i.e. a continuum (which usually has been assumed so far), or at least a dense set, or, on the contrary, a discreet set? Physicists usually call this problem the problem of the quantization of time, and formulate it as follows: is there an elementary indivisible unit of time, elementary persistence (time quantum) or, in other words, is it meaningful to introduce the smallest possible units of time? Then, there is the second (sixth) problem of the infinite divisibility of matter in temporal aspect: are there absolutely elementary (the shortest possible) processes, or can any process be divided ad infinitum into increasingly short processes? Then comes the third (seventh) problem of the infinite divisibility of space. In mathematical terms, it can be formulated as follows: is space a continuous set (as previously has usually been assumed), or at least a dense set, or, to the contrary, a discreet one? Physicists call this problem the problem of the quantization of space, which is closely connected with the problem of the quantization of time, and ask the following question: is there an elementary, indivisible unit of length (and, respectively, the unit of volume), elementary length (the quantum of space), or in other words can one introduce any units of length (or volume) which are as small as possible? The fourth (eighth) problem is the one of infinite divisibility of matter in spatial aspect: are there absolutely elementary particles, i.e. particles with no internal degrees of freedom that are indivisible, have no internal structure and no parts they could be dissected into (absolutely simple)? All these problems are clearly linked in pairs (equivalent in pairs) according to some natural and almost universally accepted physical assumption, namely that there is no empty time and no empty space (absolute vacuum). In this case, the temporal infinity of the Universe would imply a genetic infinity (for all of the infinite time would have to be “filled” with matter) and vice versa (for infinite temporal persistence of the Universe would not fit into finite time). The same holds true for spatial and material infinity, and also – or so it seems – respective pairs of the problems of “inward” infinity. If so, then only formulae are independent, at least in principle, while the solutions are closely connected in respective pairs. Theoretically, we have eight problems of the spatiotemporal infinity of the Universe but because of the equivalence of the problems in pairs we are actually facing the necessity to solve four independent problems, which, even if linked, do not appear to be related. It seems that none of these problems have been solved by physics, cosmology or philosophy until the present date. Research conducted A. Friedman, A. Einstein and others on cosmological applications of the general theory of relativity