Infinite divisibility Guide, Meaning , Facts, Information and Description
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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Atomism denies that matter is infinitely divisible.
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 phyiscal 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 is infinitely divisible if any region in space, no matter how small, can be further split. Similarly, time is infinitely divisible if any interval of time can be further split; the alternative is that time comes in discrete moments.
One dollar, or one euro, is divided into 100 cents; one cannot pay one millionth of one cent. In that sense, money 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.
To say that the field of rational numbers is infinitely divisible 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.
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 correspondes in a natural way to a Lévy process.
This concept of infinite divisibility of probability distributions was introduced in 1929 by Bruno de Finetti.A stochastic process is said to have independent increments when its non-overlapping changes are mutually independent random variables. This is an Article on Infinite divisibility. Page Contains Information, Facts Details or Explanation Guide About Infinite divisibility Philosophy
Physics
Economics
Order theory
Probability distributions