What is a finite amount

Finite amount

In set theory, a branch of mathematics, there is one finite amount a lot with a finite number of elements. Such is the amount, for example

a finite set with four elements. By definition, the empty set has no elements, i.e. the number of elements (cardinality or cardinality) is 0, so it is also considered a finite set. The cardinality of a finite set is identified with a natural number (including zero). One also writes for it | M. | = n, where the natural number n the number of elements of the set M. designated.

A set that is not finite is called an infinite set.

definition

The bijection f indicated by the red arrows shows | M. | = | M.4 | and thus the finiteness of M

A lotM. called at lastif it's a natural numbern there so that a bijection

between M and the set of all natural numbers less than n, {m | m < n} = {0,1,2,3,...,n - 1}, exists.

So in particular is for n = 0 the empty set finite, since there is a bijection between the empty set and the empty set (all natural numbers less than 0, such do not exist).

Such is the amount, for example

finally, there is a bijection to the crowd

exists, see the figure on the right.

For the set of all natural numbers

on the other hand, there is no such bijection onto one finite Crowd, the crowd is therefore infinite.

Fundamental properties of finite sets

  • Any subset of a finite set A. is also finite.
  • are A.,B. finite sets, so are their union set, their intersection and their difference amount at last. The difference amount is even finite, if only A. is finite.
  • For the cardinality of the union applies , are A. and B.disjoint, so one has .
  • The power set of a finite set has a greater power than the set itself, but it is still finite, we have | P.(A.) | = 2 | A. | .
  • The Cartesian product of finite sets is finite. Its power is greater than that of all the factors involved if no factor is empty and at least two factors have a power greater than 1. For finite sets A.,B. applies

Dedekind finiteness

Another distinction between finite and infinite sets comes from Dedekind. He defined:

  • A lot M. called at lastif it is not equal to any real subset, otherwise infinite.

Today we speak of Dedekind finiteness or. Dedekind infinity

In order to show that every finite set is also Dedekind finite, it suffices to show the following:

  1. The empty set is not equal to any real subset.
  2. If M. is not equal to any real subset, then also is not equal to any real subset (of itself).

(Point 1 is clear, since the empty set has no real subsets. For point 2 one has to show that one comes from a bijection f ' between the crowd M ' := and a real subset U ' of M ' a bijection f between M. and a real subset U can win.)

Conversely, every Dedekind is finite A. Amount also finally, because would be A. infinite, one could use the axiom of choice as a consequence of different elements in pairs Find. The image

then shows that A. to the real subset is equal in power and therefore not Dedekind finite. Contradiction!

Hereditary finite sets

A lot A. called hereditary at last, if the -transitive hull is finite. That’s not just that A. but also all elements are finite A. are finite sets, and the elements of which are also finite sets, and so on.

By definition, all hereditary finite sets are finite. The reverse is not true, so is something a finite set, because it is the only element it contains but the element itself is not finite.

In abstract set theory, the natural numbers are introduced as hereditary finite sets:

Thus the natural numbers themselves are finite sets, even hereditary finite, and we have | n | = n for every natural number n, whereby the vertical lines do not stand for the amount function, but for the thickness. That is the reason why we in the introduction of an equipotence to it instead of have spoken. The latter would have been correct, but the choice made fits the definition of natural numbers better. After that, a lot has the power nwhen to n is equal.

Averages, unions, and products of hereditary finite sets are again hereditary finite. The set of all hereditary finite sets is exactly the level V.ω the von Neumann hierarchy of the Zermelo-Fraenkel set theory.

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