Infinity
The concept of "infinity" is perhaps best approached through the concept of "the infinite." Intuitively, what is infinite has no end. If the universe is infinite--and you could travel forever in one direction, never reaching the edge of space--then the idea of "forever" requires that time, too, must be infinite.
These intuitions are of little use to physics, however, without some further mathematical precision. Mathematically, the idea of infinity is, in essence, the idea that there is no largest number (integer). For any number n that you advance as the largest number, I can counter with n + 1.
Calculus is built on the idea of infinity. Differential calculus finds the slope of a curve at an infinitely small point; integral calculus finds the area under a curve by adding up infinitely thin slivers (of the space bounded by the curve). In both cases, the properties of the infinitely small are defined by the tendencies of the very small. That is, the value of a function at infinity is defined as the limit of the function as it approaches infinity. Thus the limit of the function f(x) = 1/x as x approaches infinity is zero. As x gets very big, 1/x gets very small.
Other intuitions of infinity with mathematically interesting consequences emerge from Zeno's paradoxes. One example: to walk across a room, you must first walk across half the room. You must then cross half of the remaining half, and half of the next half, and half of the half after that, etc., ad infinitum. Though the distances to be traversed are smaller each time, they are still infinite in number, so, says Zeno, you can never reach the other side of the room.
The paradox relies on the intuition that an infinite series cannot have a finite sum; i.e., that 1/2 + 1/4 + 1/8 +...+ 1/2n etc. must add up to infinity. In fact, that intuition is wrong. As the room-crossing example might suggest (the fractions being, after all, fractions of the length of the room), the sum of the 1/2 + 1/4 + 1/8 +...+ 1/2n series is one (1). In other words, the sum or an infinite series can be a finite number.
Series that sum to a constant are said to converge, while those that sum to infinity are said to diverge--and telling them apart can be very difficult. The sums of infinite series that converge are often very useful constants. Fifty years ago, for example, cosmologist Fred Hoyle thought he had discovered such a constant and used it to defend a steady-state model of the universe (which is opposed to the currently accepted big bang model). A young graduate student named Stephen Hawking put himself on the physics map when he countered Hoyle's claim, telling Hoyle that the quantity he referred to diverges. "Of course it doesn't," replied Hoyle. "It does. I worked it out," said Hawking. Hawking was right; the series did diverge. Cosmologists' view of the universe is thus dependent upon the solution of infinite series problems.
In the first half of the twentieth century, infinity was a popular topic for philosophers. German mathematician Georg Cantor distinguished between different orders of infinity. The counting numbers, for instance, extend infinitely, but there are also infinitely many numbers (including an infinity of irrationals) between any two of them. Cantor described as "countable" any infinite set that could be placed in a one-to-one correspondence with the counting numbers, and his "diagonal proof" of the countability of the rationals was considered, by many, a great breakthrough. But these philosophical distinctions have in general proven to have little use for physicists.
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