Finite set

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In mathematics (particularly set theory), a finite set is a set that has a number of elements. Finite set_sentence_0

Informally, a finite set is a set which one could in principle count and finish counting. Finite set_sentence_1

For example, Finite set_sentence_2

is a finite set with five elements. Finite set_sentence_3

The number of elements of a finite set is a natural number (a non-negative integer) and is called the cardinality of the set. Finite set_sentence_4

A set that is not finite is called infinite. Finite set_sentence_5

For example, the set of all positive integers is infinite: Finite set_sentence_6

Finite sets are particularly important in combinatorics, the mathematical study of counting. Finite set_sentence_7

Many arguments involving finite sets rely on the pigeonhole principle, which states that there cannot exist an injective function from a larger finite set to a smaller finite set. Finite set_sentence_8

Definition and terminology Finite set_section_0

Formally, a set S is called finite if there exists a bijection Finite set_sentence_9

for some natural number n. The number n is the set's cardinality, denoted as |S|. Finite set_sentence_10

The empty set {} or Ø is considered finite, with cardinality zero. Finite set_sentence_11

If a set is finite, its elements may be written — in many ways — in a sequence: Finite set_sentence_12

In combinatorics, a finite set with n elements is sometimes called an n-set and a subset with k elements is called a k-subset. Finite set_sentence_13

For example, the set {5,6,7} is a 3-set – a finite set with three elements – and {6,7} is a 2-subset of it. Finite set_sentence_14

Basic properties Finite set_section_1

Any proper subset of a finite set S is finite and has fewer elements than S itself. Finite set_sentence_15

As a consequence, there cannot exist a bijection between a finite set S and a proper subset of S. Any set with this property is called Dedekind-finite. Finite set_sentence_16

Using the standard ZFC axioms for set theory, every Dedekind-finite set is also finite, but this implication cannot be proved in ZF (Zermelo–Fraenkel axioms without the axiom of choice) alone. Finite set_sentence_17

The axiom of countable choice, a weak version of the axiom of choice, is sufficient to prove this equivalence. Finite set_sentence_18

Any injective function between two finite sets of the same cardinality is also a surjective function (a surjection). Finite set_sentence_19

Similarly, any surjection between two finite sets of the same cardinality is also an injection. Finite set_sentence_20

The union of two finite sets is finite, with Finite set_sentence_21

In fact, by the inclusion–exclusion principle: Finite set_sentence_22

More generally, the union of any finite number of finite sets is finite. Finite set_sentence_23

The Cartesian product of finite sets is also finite, with: Finite set_sentence_24

Similarly, the Cartesian product of finitely many finite sets is finite. Finite set_sentence_25

A finite set with n elements has 2 distinct subsets. Finite set_sentence_26

That is, the power set of a finite set is finite, with cardinality 2. Finite set_sentence_27

Any subset of a finite set is finite. Finite set_sentence_28

The set of values of a function when applied to elements of a finite set is finite. Finite set_sentence_29

All finite sets are countable, but not all countable sets are finite. Finite set_sentence_30

(Some authors, however, use "countable" to mean "countably infinite", so do not consider finite sets to be countable.) Finite set_sentence_31

The free semilattice over a finite set is the set of its non-empty subsets, with the join operation being given by set union. Finite set_sentence_32

Necessary and sufficient conditions for finiteness Finite set_section_2

In Zermelo–Fraenkel set theory without the axiom of choice (ZF), the following conditions are all equivalent: Finite set_sentence_33

Finite set_ordered_list_0

  1. S is a finite set. That is, S can be placed into a one-to-one correspondence with the set of those natural numbers less than some specific natural number.Finite set_item_0_0
  2. (Kazimierz Kuratowski) S has all properties which can be proved by mathematical induction beginning with the empty set and adding one new element at a time. (See below for the set-theoretical formulation of Kuratowski finiteness.)Finite set_item_0_1
  3. (Paul Stäckel) S can be given a total ordering which is well-ordered both forwards and backwards. That is, every non-empty subset of S has both a least and a greatest element in the subset.Finite set_item_0_2
  4. Every one-to-one function from P(P(S)) into itself is onto. That is, the powerset of the powerset of S is Dedekind-finite (see below).Finite set_item_0_3
  5. Every surjective function from P(P(S)) onto itself is one-to-one.Finite set_item_0_4
  6. (Alfred Tarski) Every non-empty family of subsets of S has a minimal element with respect to inclusion. (Equivalently, every non-empty family of subsets of S has a maximal element with respect to inclusion.)Finite set_item_0_5
  7. S can be well-ordered and any two well-orderings on it are order isomorphic. In other words, the well-orderings on S have exactly one order type.Finite set_item_0_6

If the axiom of choice is also assumed (the axiom of countable choice is sufficient), then the following conditions are all equivalent: Finite set_sentence_34

Finite set_ordered_list_1

  1. S is a finite set.Finite set_item_1_7
  2. (Richard Dedekind) Every one-to-one function from S into itself is onto.Finite set_item_1_8
  3. Every surjective function from S onto itself is one-to-one.Finite set_item_1_9
  4. S is empty or every partial ordering of S contains a maximal element.Finite set_item_1_10

Foundational issues Finite set_section_3

Georg Cantor initiated his theory of sets in order to provide a mathematical treatment of infinite sets. Finite set_sentence_35

Thus the distinction between the finite and the infinite lies at the core of set theory. Finite set_sentence_36

Certain foundationalists, the strict finitists, reject the existence of infinite sets and thus recommend a mathematics based solely on finite sets. Finite set_sentence_37

Mainstream mathematicians consider strict finitism too confining, but acknowledge its relative consistency: the universe of hereditarily finite sets constitutes a model of Zermelo–Fraenkel set theory with the axiom of infinity replaced by its negation. Finite set_sentence_38

Even for those mathematicians who embrace infinite sets, in certain important contexts, the formal distinction between the finite and the infinite can remain a delicate matter. Finite set_sentence_39

The difficulty stems from Gödel's incompleteness theorems. Finite set_sentence_40

One can interpret the theory of hereditarily finite sets within Peano arithmetic (and certainly also vice versa), so the incompleteness of the theory of Peano arithmetic implies that of the theory of hereditarily finite sets. Finite set_sentence_41

In particular, there exists a plethora of so-called non-standard models of both theories. Finite set_sentence_42

A seeming paradox is that there are non-standard models of the theory of hereditarily finite sets which contain infinite sets, but these infinite sets look finite from within the model. Finite set_sentence_43

(This can happen when the model lacks the sets or functions necessary to witness the infinitude of these sets.) Finite set_sentence_44

On account of the incompleteness theorems, no first-order predicate, nor even any recursive scheme of first-order predicates, can characterize the standard part of all such models. Finite set_sentence_45

So, at least from the point of view of first-order logic, one can only hope to describe finiteness approximately. Finite set_sentence_46

More generally, informal notions like set, and particularly finite set, may receive interpretations across a range of formal systems varying in their axiomatics and logical apparatus. Finite set_sentence_47

The best known axiomatic set theories include Zermelo-Fraenkel set theory (ZF), Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC), Von Neumann–Bernays–Gödel set theory (NBG), Non-well-founded set theory, Bertrand Russell's Type theory and all the theories of their various models. Finite set_sentence_48

One may also choose among classical first-order logic, various higher-order logics and intuitionistic logic. Finite set_sentence_49

A formalist might see the meaning of set varying from system to system. Finite set_sentence_50

Some kinds of Platonists might view particular formal systems as approximating an underlying reality. Finite set_sentence_51

Set-theoretic definitions of finiteness Finite set_section_4

Various properties that single out the finite sets among all sets in the theory ZFC turn out logically inequivalent in weaker systems such as ZF or intuitionistic set theories. Finite set_sentence_52

Two definitions feature prominently in the literature, one due to Richard Dedekind, the other to Kazimierz Kuratowski. Finite set_sentence_53

(Kuratowski's is the definition used above.) Finite set_sentence_54

Kuratowski finiteness is defined as follows. Finite set_sentence_55

Given any set S, the binary operation of union endows the powerset P(S) with the structure of a semilattice. Finite set_sentence_56

Writing K(S) for the sub-semilattice generated by the empty set and the singletons, call set S Kuratowski finite if S itself belongs to K(S). Finite set_sentence_57

Intuitively, K(S) consists of the finite subsets of S. Crucially, one does not need induction, recursion or a definition of natural numbers to define generated by since one may obtain K(S) simply by taking the intersection of all sub-semilattices containing the empty set and the singletons. Finite set_sentence_58

Readers unfamiliar with semilattices and other notions of abstract algebra may prefer an entirely elementary formulation. Finite set_sentence_59

Kuratowski finite means S lies in the set K(S), constructed as follows. Finite set_sentence_60

Write M for the set of all subsets X of P(S) such that: Finite set_sentence_61

Finite set_unordered_list_2

  • X contains the empty set;Finite set_item_2_11
  • For every set T in P(S), if X contains T then X also contains the union of T with any singleton.Finite set_item_2_12

Then K(S) may be defined as the intersection of M. Finite set_sentence_62

In ZF, Kuratowski finite implies Dedekind finite, but not vice versa. Finite set_sentence_63

In the parlance of a popular pedagogical formulation, when the axiom of choice fails badly, one may have an infinite family of socks with no way to choose one sock from more than finitely many of the pairs. Finite set_sentence_64

That would make the set of such socks Dedekind finite: there can be no infinite sequence of socks, because such a sequence would allow a choice of one sock for infinitely many pairs by choosing the first sock in the sequence. Finite set_sentence_65

However, Kuratowski finiteness would fail for the same set of socks. Finite set_sentence_66

Other concepts of finiteness Finite set_section_5

In ZF set theory without the axiom of choice, the following concepts of finiteness for a set S are distinct. Finite set_sentence_67

They are arranged in strictly decreasing order of strength, i.e. if a set S meets a criterion in the list then it meets all of the following criteria. Finite set_sentence_68

In the absence of the axiom of choice the reverse implications are all unprovable, but if the axiom of choice is assumed then all of these concepts are equivalent. Finite set_sentence_69

(Note that none of these definitions need the set of finite ordinal numbers to be defined first; they are all pure "set-theoretic" definitions in terms of the equality and membership relations, not involving ω.) Finite set_sentence_70

Finite set_unordered_list_3

  • I-finite. Every non-empty set of subsets of S has a ⊆-maximal element. (This is equivalent to requiring the existence of a ⊆-minimal element. It is also equivalent to the standard numerical concept of finiteness.)Finite set_item_3_13
  • Ia-finite. For every partition of S into two sets, at least one of the two sets is I-finite.Finite set_item_3_14
  • II-finite. Every non-empty ⊆-monotone set of subsets of S has a ⊆-maximal element.Finite set_item_3_15
  • III-finite. The power set P(S) is Dedekind finite.Finite set_item_3_16
  • IV-finite. S is Dedekind finite.Finite set_item_3_17
  • V-finite. ∣S∣ = 0 or 2⋅∣S∣ > ∣S|.Finite set_item_3_18
  • VI-finite. ∣S∣ = 0 or ∣S∣ = 1 or ∣S∣ > ∣S∣.Finite set_item_3_19
  • VII-finite. S is I-finite or not well-orderable.Finite set_item_3_20

The forward implications (from strong to weak) are theorems within ZF. Finite set_sentence_71

Counter-examples to the reverse implications (from weak to strong) in ZF with urelements are found using model theory. Finite set_sentence_72

Most of these finiteness definitions and their names are attributed to by , p. 278. Finite set_sentence_73

However, definitions I, II, III, IV and V were presented in , pp. 49, 93, together with proofs (or references to proofs) for the forward implications. Finite set_sentence_74

At that time, model theory was not sufficiently advanced to find the counter-examples. Finite set_sentence_75

Each of the properties I-finite thru IV-finite is a notion of smallness in the sense that any subset of a set with such a property will also have the property. Finite set_sentence_76

This is not true for V-finite thru VII-finite because they may have countably infinite subsets. Finite set_sentence_77

See also Finite set_section_6

Finite set_unordered_list_4

Credits to the contents of this page go to the authors of the corresponding Wikipedia page: set.