Infinite divisibility
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The concept of infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter, space, time, money, or abstract mathematical objects.
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Philosophy
Atomism denies that matter is infinitely divisible.
Physics
In physics, the question of whether matter is infinitely divisible is the question of whether it is true that no matter how small the pieces into which a physical object has been cut, they can be split further. The word atom originally meant a smallest possible particle of matter, which cannot be further divided. Later, those objects to which the name atom had been assigned were found to be further divisible, but the word atom nonetheless continues to refer to them.
Physical space has often been regarded as infinitely divisible: it was thought that any region in space, no matter how small, could be further split. Similarly, time was regarded as infinitely divisible.
However, the pioneering work of Max Planck (1858–1947) in the field of quantum physics suggests that there is, in fact, a minimum distance (now called the Planck length, 1.616 × 10−35 metres) and therefore a minimum time interval (the amount of time which light takes to traverse that distance in a vacuum, 5.391 × 10−44 seconds, known as the Planck time) smaller than which meaningful measurement is impossible.
Economics
One dollar, or one euro, is divided into 100 cents; one can only pay in increments of a cent. It is quite commonplace for prices of some commodities such as gasoline to be in increments of a tenth of a cent per gallon or per litre. However, payment is not infinitely divisible.
Although time may be infinitely divisible, data on securities prices are reported at discrete times. For example, if one looks at records of stock prices in the 1920s, one may find the prices at the end of each day, but perhaps not at three-hundredths of a second after 12:47 PM. Thus time in market records is not infinitely divisible. Perhaps paradoxically, technical mathematics applied to financial markets is often simpler if infinitely divisible time is used as an approximation.
Order theory
To say that the field of rational numbers is infinitely divisible (i.e. order theoretically dense) means that between any two rational numbers there is another rational number. By contrast, the ring of integers is not infinitely divisible.
Infinite divisibility does not imply gap-less-ness: the rationals do not enjoy the least upper bound property. That means that one may partition the rationals into two non-empty sets A and B in such a way that every member of A is less than every member of B, and A has no largest member, and B has no smallest member. The field of real numbers, by contrast, is both infinitely divisible and gapless. Any linearly ordered set that is infinitely divisible and gapless, and has more than one member, is uncountably infinite. For a proof, see Cantor's first uncountability proof. Infinite divisibility alone implies infiniteness but not uncountability, as the rational numbers exemplify.
Probability distributions
To say that a probability distribution F on the real line is infinitely divisible means that if X is any random variable whose distribution is F, then for every positive integer n there exist n independent identically distributed random variables X1, ..., Xn whose sum is X (those n other random variables do not usually have the same probability distribution that X has (but do sometimes, as in the case of the Cauchy distribution)).
The Poisson distributions, the normal distributions, and the gamma distributions are infinitely divisible probability distributions.
Every infinitely divisible probability distribution corresponds in a natural way to a Lévy process, i.e., a stochastic process { Xt : t ≥ 0 } with stationary independent increments (stationary means that for s < t, the probability distribution of Xt − Xs depends only on t − s; independent increments means that that difference is independent of the corresponding difference on any interval not overlapping with [s, t], and similarly for any finite number of intervals).
This concept of infinite divisibility of probability distributions was introduced in 1929 by Bruno de Finetti.