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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 $S$ .

finite is owned by David Jao.

How to Cite This Entry

David Jao. "finite" (version 4). PlanetMath.org. Freely available at http://planetmath.org/Finite.html.

Classification

AMS MSC:

03E10 (Mathematical logic and foundations :: Set theory :: Ordinal and cardinal numbers)

Pending Errata and Addenda

None. [ View all 4 ]

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Discussion

Tarski's definition of a finite set by smw on 2006-02-20 20:09:15

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

[ reply | up ]

Re: Tarski's definition of a finite set by ratboy on 2006-02-22 12:40:59

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