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: ${\displaystyle M}$

Every non-empty subset system contains a with the property that no real subset still applies. ${\displaystyle {\mathcal {T}}\subseteq {\mathcal {P}}(M)}$ ${\displaystyle T\in {\mathcal {T}}}$ ${\displaystyle S\subsetneqq T}$ ${\displaystyle S\in {\mathcal {T}}}$

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 ) . ${\displaystyle {\mathcal {T}}}$ ${\displaystyle {\mathcal {T}}}$

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 ) . ${\displaystyle {\mathcal {T}}}$ ${\displaystyle {\mathcal {T}}}$

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 . ${\displaystyle M}$ ${\displaystyle \emptyset \neq {\mathcal {T}}\subseteq {\mathcal {P}}(M)}$ ${\displaystyle \{|T|\mid T\in {\mathcal {T}}\}\subseteq \mathbb {N} }$ ${\displaystyle \mathbb {N} }$ ${\displaystyle n}$ ${\displaystyle n=|T_{0}|}$ ${\displaystyle T_{0}\in {\mathcal {T}}}$ ${\displaystyle T_{0}}$ ${\displaystyle {\mathcal {T}}}$

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 . ${\displaystyle M}$ ${\displaystyle {\mathcal {T}}}$ ${\displaystyle T\subseteq M}$ ${\displaystyle {\mathcal {T}}}$ ${\displaystyle M\in {\mathcal {T}}}$

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. ${\displaystyle T\in {\mathcal {T}}}$ ${\displaystyle T\in {\mathcal {T}}}$ ${\displaystyle t\in T}$ ${\displaystyle t}$ ${\displaystyle T\setminus \{t\}}$ ${\displaystyle T}$ ${\displaystyle T}$ ${\displaystyle t}$ ${\displaystyle T\in {\mathcal {T}}}$ ${\displaystyle {\mathcal {T}}}$

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

• Finite amount
• Infinite amount
• infinity

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

1. ^ Tarski: Fundamenta Mathematicae . tape 6 , p. 45 ff .
2. ↑ Sierpiński: p. 50 ff.
3. ↑ Felscher: p. 180 ff.
4. ↑ Jech: p. 14.
5. ↑ Lüneburg: pp. 52–54.
6. ↑ Felscher: p. 181.
7. ↑ Felscher: p. 181.
8. ↑ Sierpiński: p. 50.
9. ↑ Jech: p. 14.
10. ↑ Felscher: p. 175.
11. ↑ Felscher: pp. 175–177.
12. ↑ Felscher: pp. 175-185.
13. ↑ Sierpiński: p. 50.