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Finite sets

There are several ways how to define finite sets. The definition is usually based on the fact that assuming Axiom of Choice , the following conditions are all equivalent for a set :

1. there is an injective (i.e. one-to-one) correspondence between the set and some set of the form , where is a non-negative integer. The case covers the empty set , which is considered finite
2. (Paul Stäckel)   can endowed with total ordering in which every non-empty subset of has both a least and a greatest element in the subset
3. (Richard Dedekind)  Every function from S one-to-one into itself is onto, that is there exists no bijection between the set and any of its proper subsets
4. (Alfred Tarski) Every non-empty subset of the power set of has a minimal element with respect to inclusion.
5. Every non-empty subset of the power set of has a maximal element with respect to the set inclusion.
6. Every function from onto itself is one-to-one
7. can endowed with a well-ordered and any two well-orderings on are order isomorphic, that is the well-orderings on have exactly one order type.

For instance, a set satisfying property (iii) is called Dedekind finite, in honor of R.Dedekind who, as indicated above, used this property to define finite sets  [1] , § 64-70 (actually he firstly  defined the concept of an infinite set , and then he says that a set is finite if it is not  infinite).

Theorem: Assuming the Axiom of Choice   a set is finite in sense of (i) if and only if it is Dedekind finite.

Proposition: If , are finite sets the so is .

If there exists a bijection between and then is said to be of cardinality and we write or that .

Proposition: If and are finite sets then .

Proof. We prove the result if . Let , and , be the bijections form the definition of the cardinality. Define by

to get the required bijection.

References

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