# Chapter 1: Spacetime

## Infinity

It appears, then, that the assumption of nothing implies an infinite, four-dimensional expanse of space and time (not yet spacetime, which is significantly different)— still nothing, to be sure, but at least a somewhat more interesting version of it. To take another step toward existence, it is necessary to examine this curious notion of infinity that is inextricably bound to the assumption of nothing.

The first thing to notice is that infinity is an inherently irrational concept. Though we may understand in a strictly formal sense what the word infinity means, it is not possible to conjure up an accurate representation of the idea in our minds. The best we can do is acknowledge that, however far we go, we can always go farther. But man can’t wrap his head around anything truly boundless. Moreover, the machinery of logical and mathematical reasoning also breaks down when applied to infinity. The crux of this breakdown comes from the observation that the cardinality (size) of all infinite sets is the same, regardless of how those sets are defined. For example, the set of all integers is the same size as the set of all odd numbers, even though, intuitively, it seems like there should be twice as many of the former as the latter. The even numbers are missing from the set of odd numbers, but not missing from the set of integers. Therefore, the set of integers must, in some sense, be the larger of the two, even if we concede that both are infinite. But how could one infinite set be any larger than another? They both go on forever.

Any number of paradoxes can be formulated by applying the above observation to hypothetical situations. David Hilbert’s paradox of the Infinite Hotel is one example. In it, we are to imagine a hotel with an infinite number of rooms, and then wrestle with various notions of vacancy and occupancy. Specifically, would an infinite number of guests result in full occupancy? The answer appears to be no. If a new guest arrives, we simply move the guest in room one to room two, the guest in room two to room three, and so on, making room for the new guest. Since there is no end to the number of rooms, even an infinite number of guests cannot fill them all. In this and every other paradox of infinity, the issue revolves around treating infinity simultaneously as a number and as the concept of unboundedness. A number is a discrete, definable entity, while unboundedness is exactly the opposite. All numbers are unique, their values rigorously determined, whereas all unboundedness, qua infinity, is the same. But because we can define infinite sets in much the same way that we define particular numbers, it appears as though different infinities are equal and unequal at the same time.

These sorts of paradoxes are interesting, but they are only relevant outside of pure mathematics if there are, in fact, genuine infinities in the physical world. Currently, infinities are rejected by physicists as meaningless, and none of the accepted laws of nature require them. On the contrary, an infinite answer to an equation describing a physical phenomenon is regarded as evidence of a mistake. Consider, if there were any infinite physical quantities, they would, by definition, take over the entire cosmos. Infinite gravity would pull everything in with an infinite force. An infinite force would generate an infinite quantity of energy. Infinite energy, in turn, would impart an infinite expansive or implosive velocity to everything in the universe. Nothing in our experience justifies these crazy conclusions, hence infinity is never relevant or even possible in the real world.

Yet infinite nothingness appears inescapable. And, as with the paradoxes discussed above, it is easy to construct a contradiction between the finite character of any discrete region of the void, and its infinite character, taken as a whole. Imagine, for example, an infinite spherical region of the void (fig. 1); being infinite, the void can contain any number of infinite sub-regions, just as we can define any number of infinite sets using only a subset of the integers (odd numbers, for example). Now, any discrete point selected anywhere inside of this infinite sphere is, by definition, an infinite distance from the perimeter. And because all infinite quantities are equal (equally boundless), every point in the sphere is also an equal distance from the perimeter. However, the only point in a sphere that is equidistant from every point on the perimeter is the very center of the sphere. Therefore, the line connecting any point inside the sphere to its perimeter is a radius of that sphere. That is, every point in the sphere, no matter where it is, is the same point, namely, the center. The paradox is obvious—every point in an infinite sphere is the center of the sphere, the same point. There is a clear logical contradiction between infinite geometry and Euclidean geometry. Indeed, there is a contradiction between infinity and every variety of math and logic, because every infinity must be treated both as a particular number as well as an equally unbounded quantity. Or again, infinity can be defined in many different (and mutually exclusive) ways, but always ends up equally infinite just the same.

Figure 1—While only one point (the center) is equidistant from every point on the perimeter of a finite, Euclidean sphere (a) every point in the interior of an infinite sphere (b) is equidistant from the perimeter. Hence, every point in an infinite sphere can be thought of as its center.

The above paradox is even clearer if we create a simple isosceles triangle (fig. 2) and vary the height. As the height increases, the angle at a decreases, and if the height becomes infinite, the angle becomes zero. However, if this angle becomes zero, points b and c become the same point. This is true regardless of how far apart, in absolute terms, b and c really are. That is, b and c, from the standpoint of infinity, are the same point, even though they aren’t really the same point. Under normal, finite conditions, these sorts of paradoxes are no more than interesting intellectual observations, having no relationship to reality. But, if we are agreed that the void is genuinely and unavoidably infinite, we can’t simply leave this problem unaddressed. The points in an infinite sphere are either all in the center or they’re not. Points b and c either have a particular separation or they don’t. The void is either infinite or it isn’t. In none of these examples can we have it both ways.

Figure 2—As long as the height of the triangle is finite, the angle at a is greater than zero and the points b and c have a positive separation. But when the height becomes infinite, the angle goes to zero and points b and c become the same point.

From an intuitive perspective, we might try to resolve this matter by pointing out that, with infinite distances at our disposal, it is always possible to stand back from an object, however big it might be, far enough to reduce it to a pin point. Venus, for example, looks to the naked eye like a point, but only because it is so far away. If we launch a space probe to get a closer look, its true size becomes evident. There is no paradox to unravel. But though this might seem to resolve the issue, it ignores the categorical difference between really, really big, on the one hand, and infinite, on the other. As we increase the height of our triangle, the distance of a from b and c is not merely great enough to make b and c look the same, it is great enough to render them, mathematically, as the exact same point. The angle at a, from an infinite distance, is not just very, very small, it is exactly zero. And this is true whether we initially choose the base to be an inch or a light year wide. This results in a real, intractable mathematical contradiction. There appears to be a kind of tension between the Euclidean and infinite characters of the points b and c. The question now is, do we treat this tension as entirely theoretical, or is it, in some sense, real?