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# finite

A set $S$ is *finite* if there exists a natural number $n$ and a bijection from $S$ to $n$. Note that we are using the set theoretic definition of natural number, under which the natural number $n$ equals the set $\{0,1,2,\ldots,n-1\}$. If there exists such an $n$, then it is unique, and we call $n$ the *cardinality* of $S$.

Equivalently, a set $S$ is finite if and only if there is no bijection between $S$ and any proper subset of $S$.

Defines:

finite set

Related:

Infinite

Type of Math Object:

Definition

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Reference

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## Mathematics Subject Classification

03E10*no label found*92C05

*no label found*92B05

*no label found*18-00

*no label found*92C40

*no label found*18-02

*no label found*

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## Comments

## Tarski's definition of a finite set

According to Suppes [p. 100], Tarski proposed a slick definition of finiteness which mentions neither the natural numbers nor bijections. Here it is (quoting Suppes):

``A is finite if and only if every non-empty family of subsets of A has a minimal element. ''

In this context, a minimal element in a family F of subsets is an element x in F such that there is no y in F such that y is a *proper* subset of x.

Granted, at first glace, this definition is not intuitively clear to most people (i.e. non set-theorists). But one can probably get used to it after a while. This is the formal definition that Suppes adopts in his text.

Here's a cool fact: Unlike Dedekind's definition (i.e. a set is finite if and only if it is not bijective to a proper subset), ``Tarski's definition does not require the axiom of choice to prove its equivalence to the ordinary numerical definition." [Suppes, p. 99]

According to Suppes, Tarski proposed this definition in his paper

``Sur les ensembles finis," published in _Fundamenta_Mathematicae_, Vol 6 (1924b), pp. 45 - 95. This would be an interesting paper to read. Not only does Tarski propose a new definition, he also gives a complete survey of all of the ``non-numerical definitions of finitude" up to that time.

Reference:

Suppes, P. _Axiomatic_Set_Theory_, Dover, New York: 1972

## Re: Tarski's definition of a finite set

> According to Suppes, Tarski proposed this definition in his

> paper

> ``Sur les ensembles finis," published in

> _Fundamenta_Mathematicae_, Vol 6 (1924b), pp. 45 - 95. This

> would be an interesting paper to read. Not only does Tarski

> propose a new definition, he also gives a complete survey of

> all of the ``non-numerical definitions of finitude" up to

> that time.

Available for download: http://matwbn.icm.edu.pl/tresc.php?wyd=1&tom=6