Editing Finite set (section)

== Necessary and sufficient conditions for finiteness ==
{{anchor|Tarski finite}}

In [[Zermelo–Fraenkel set theory]] without the axiom of choice (ZF), the following conditions are all equivalent:<ref>{{Cite web |title=Art of Problem Solving |url=https://artofproblemsolving.com/wiki/index.php/Zermelo-Fraenkel_Axioms |access-date=2022-09-07 |website=artofproblemsolving.com}}</ref>

# <math>S</math> is a finite set. That is, <math>S</math> can be placed into a one-to-one correspondence with the set of those natural numbers less than some specific natural number.
# ([[Kazimierz Kuratowski]]) <math>S</math> has all properties which can be proved by mathematical induction beginning with the empty set and adding one new element at a time.
# ([[Paul Stäckel]]) <math>S</math> can be given a [[total order]]ing which is [[well-order]]ed both forwards and backwards. That is, every non-empty subset of <math>S</math> has both a least and a greatest element in the subset.
# Every one-to-one function from <math>\wp\bigl(\wp(S)\bigr)</math> into itself is [[onto]]. That is, the [[powerset]] of the powerset of <math>S</math> is Dedekind-finite (see below).<ref>The equivalence of the standard numerical definition of finite sets to the Dedekind-finiteness of the power set of the power set was shown in 1912 by {{harvnb|Whitehead|Russell|2009|p=288}}. This Whitehead/Russell theorem is described in more modern language by {{harvnb|Tarski|1924|pp=73–74}}.</ref>
# Every surjective function from <math>\wp\bigl(\wp(S)\bigr)</math> onto itself is one-to-one.
# ([[Alfred Tarski]]) Every non-empty family of subsets of <math>S</math> has a [[minimal element]] with respect to inclusion.<ref>{{harvnb|Tarski|1924|pp=48–58}}, demonstrated that his definition (which is also known as I-finite) is equivalent to Kuratowski's set-theoretical definition, which he then noted is equivalent to the standard numerical definition via the proof by {{harvnb|Kuratowski|1920|pp=130–131}}.</ref> (Equivalently, every non-empty family of subsets of <math>S</math> has a [[maximal element]] with respect to inclusion.)
# <math>S</math> can be well-ordered and any two well-orderings on it are [[order isomorphic]]. In other words, the well-orderings on <math>S</math> have exactly one [[order type]].

If the [[axiom of choice]] is also assumed (the [[axiom of countable choice]] is sufficient),<ref>{{cite book |last1=Herrlich |first1=Horst |title=Axiom of Choice |date=2006 |publisher=Springer |series=Lecture Notes in Mathematics |volume=1876 |doi=10.1007/11601562 |isbn=3-540-30989-6 |url=https://link.springer.com/book/10.1007/11601562 |access-date=18 July 2023|contribution=Proposition 4.13|page=48}}</ref> then the following conditions are all equivalent:

# <math>S</math> is a finite set.
# ([[Richard Dedekind]]) Every one-to-one function from <math>S</math> into itself is onto. A set with this property is called [[Dedekind-infinite set|Dedekind-finite]].
# Every surjective function from <math>S</math> onto itself is one-to-one.
# <math>S</math> is empty or every [[partial ordering]] of <math>S</math> contains a [[maximal element]].

=== Other concepts of finiteness ===

In ZF set theory without the [[axiom of choice]], the following concepts of finiteness for a set <math>S</math> are distinct. They are arranged in strictly decreasing order of strength, i.e. if a set <math>S</math> meets a criterion in the list then it meets all of the following criteria. 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.<ref>This list of 8 finiteness concepts is presented with this numbering scheme by both {{harvnb|Howard|Rubin|1998|pp=278–280}}, and {{harvnb|Lévy|1958|pp=2–3}}, although the details of the presentation of the definitions differ in some respects which do not affect the meanings of the concepts.</ref> (Note that none of these definitions need the set of finite [[ordinal number]]s to be defined first; they are all pure "set-theoretic" definitions in terms of the equality and membership relations, not involving ω.)

* '''I-finite'''. Every non-empty set of subsets of <math>S</math> has a <math>\subseteq</math>-maximal element. (This is equivalent to requiring the existence of a <math>\subseteq</math>-minimal element. It is also equivalent to the standard numerical concept of finiteness.)
* '''Ia-finite'''. For every partition of <math>S</math> into two sets, at least one of the two sets is I-finite. (A set with this property which is not I-finite is called an [[amorphous set]].<ref>{{harvtxt|de la Cruz|Dzhafarov|Hall|2006|p=8}}</ref>)
* '''II-finite'''. Every non-empty <math>\subseteq</math>-monotone set of subsets of <math>S</math> has a <math>\subseteq</math>-maximal element.
* '''III-finite'''. The power set <math>\wp(S)</math> is Dedekind finite.
* '''IV-finite'''. <math>S</math> is Dedekind finite.
* '''V-finite'''. <math>|S|=0</math> or <math>2\cdot|S|>|S|</math>.
* '''VI-finite'''. <math>|S|=0</math> or <math>|S|=1</math> or <math>|S|^2>|S|</math>.
* '''VII-finite'''. <math>S</math> is I-finite or not well-orderable.

The forward implications (from strong to weak) are theorems within ZF. Counter-examples to the reverse implications (from weak to strong) in ZF with [[urelement]]s are found using [[model theory]].<ref>{{harvnb|Lévy|1958}} found counter-examples to each of the reverse implications in Mostowski models. Lévy attributes most of the results to earlier papers by Mostowski and Lindenbaum.</ref>

Most of these finiteness definitions and their names are attributed to {{harvnb|Tarski|1954}} by {{harvnb|Howard|Rubin|1998|p=278}}. However, definitions I, II, III, IV and V were presented in {{harvnb|Tarski|1924|pp=49, 93}}, together with proofs (or references to proofs) for the forward implications. At that time, model theory was not sufficiently advanced to find the counter-examples.

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