# Tarski finiteness

The concept of **Tarski finitude** is a concept of set theory . It goes back to Alfred Tarski and his work *Sur les ensembles finis* from 1924. With the concept of *Tarski finiteness* it is possible to grasp the concept of the finite set without resorting to the natural numbers .

## definition

An amount is *finally within the meaning of Tarski* or (short) *Tarski finally* (engl. *Tarski finite* ) if it satisfies the following condition:

- Every non-empty subset system contains a with the property that no real subset still applies.

In other words, this means:

- A set is
*Tarski finite*if and only if every non-empty subset system formed from subsets of this set contains a minimal element (with regard to the subset relation to ) .

The following marking is synonymous with this:

- A set is
*Tarski finite*if and only if every non-empty subset system formed from subsets of this set contains a maximal element (with regard to the subset relation to ) .

## sentence

The following fundamental theorem shows that the concept of *Tarski* finitude leads to the usual concept of finitude :

- A lot is
*Tarski-finite*exactly when it is finite in the usual sense .

### Evidence sketch

*Tarski finiteness*follows from*finitude in the usual sense*

If one takes a finite set in the usual sense and in addition , then the non-empty set of the finite numbers formed from this subset system can be formed . Since it is well-ordered, there is a very small number in it , for example for a certain amount . This is then certainly also minimal with regard to the subset relation .

*Finiteness in the usual sense*follows from*Tarski finiteness*

If, on the other hand, there is a set that is not finite in the usual sense, then the subset system of *all can be* formed which has this property, i.e. is not finite in the usual sense. is not empty, because it applies in any case .

In addition, each can be reduced by at least one element, because such an element is not the empty set and therefore contains at least one . If this element is removed , the remaining amount is still a non-finite amount in the usual sense; because if the remainder were finite in the usual sense, this would also be itself, since the remainder can be formed by adding the element . Consequently, none of the subsets relation can be minimal.

## Further approaches to the concept of finitude

The mathematician Paul Stäckel has an approach related to the Tarskian to formulate a finite concept of sets, which is also based on concepts of order . This approach is based on the following definition of *finitude* :

- A set is
*finite in the sense of Stäckel*if and only if a total order exists on it , so that every non-empty subset contains a smallest element and a largest element with respect to this total order .

Stäckel's concept of finitude can also be described as follows:

- A set is
*finite in the sense of Stäckel*if and only if a total order exists on it , so that both this and the associated dual order relation are well orders .

The following also applies to Stäckel's concept of finitude:

- A set is
*finite in the sense of Stäckel*exactly when it is finite in the usual sense.

In addition to the Tarsky and Stäckelian approaches to the concept of the finite set, there are a number of other approaches. The concept of *finite set in the sense of Dedekind* ( *Dedekind finiteness* ) is outstanding in mathematics history . Unlike (for example) the Tarskian concept of finitude , the axiom of choice is required to prove that *Dedekind finitude* and *finitude* coincide *in the usual sense* .

## See also

## literature

- Walter Felscher : Naive quantities and abstract numbers . BI Wissenschaftsverlag, Mannheim (among others) 1978, ISBN 3-411-01538-1 ( MR0516505 ).
- Thomas Jech : Set Theory . The Third Millennium edition, revised and expanded (= Springer Monographs in Mathematics ). Springer Verlag, Berlin et al. 2003, ISBN 3-540-44085-2 ( MR1940513 ).
- Heinz Lüneburg : Tools and Fundamental Constructions of Combinatorial Mathematics . BI Wissenschaftsverlag, Mannheim (ua) 1989, ISBN 3-411-03194-8 ( MR1116324 ).
- Wacław Sierpiński : Cardinal and Ordinal Numbers . Panstwowe Wydawnictwo Naukowe, Warszawa 1958 ( MR0095787 ).
- Paul Stäckel: On H. Weber's elementary set theory . In: Jahresber. DMV . tape 16 , 1907, pp. 425-428 ( digizeitschriften.de [PDF]).
- Alfred Tarski: Sur les ensembles finis . In: Fund. Math . tape 6 , 1924, pp. 45-95 ( matwbn.icm.edu.pl [PDF]).

## References and footnotes

- ^ Tarski: Fundamenta Mathematicae . tape 6 , p. 45 ff .
- ↑ Sierpiński: p. 50 ff.
- ↑ Felscher: p. 180 ff.
- ↑ Jech: p. 14.
- ↑ Lüneburg: pp. 52–54.
- ↑ Felscher: p. 181.
- ↑ Felscher: p. 181.
- ↑ Sierpiński: p. 50.
- ↑ Jech: p. 14.
- ↑ Felscher: p. 175.
- ↑ Felscher: pp. 175–177.
- ↑ Felscher: pp. 175-185.
- ↑ Sierpiński: p. 50.