TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
(particularly
set theory Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that are relevant to ...
), a finite set is a set that has a
finite Finite is the opposite of Infinity, infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected ...
number of
element Element may refer to: Science * Chemical element Image:Simple Periodic Table Chart-blocks.svg, 400px, Periodic table, The periodic table of the chemical elements In chemistry, an element is a pure substance consisting only of atoms that all ...
s. Informally, a finite set is a set which one could in principle count and finish counting. For example, :$\$ is a finite set with five elements. The number of elements of a finite set is a
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
(a
non-negative In mathematics, the sign of a real number is its property of being either positive, negative number, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third ...
integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...
) and is called the ''
cardinality In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
'' of the set. A set that is not finite is called ''infinite''. For example, the set of all positive integers is infinite: :$\.$ Finite sets are particularly important in
combinatorics Combinatorics is an area of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geom ...
, the mathematical study of
counting Counting is the process of determining the number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) ...
. Many arguments involving finite sets rely on the
pigeonhole principle In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, which states that there cannot exist an
injective function In , an injective function (also known as injection, or one-to-one function) is a that maps elements to distinct elements; that is, implies . In other words, every element of the function's is the of one element of its . The term must no ... from a larger finite set to a smaller finite set.

# Definition and terminology

Formally, a set is called finite if there exists a
bijection In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ... :$f\colon S\to\$ for some natural number . The number is the set's cardinality, denoted as . The
empty set #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ... or ∅ is considered finite, with cardinality zero. If a set is finite, its elements may be written — in many ways — in a
sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ... : :$x_1,x_2,\ldots,x_n \quad \left(x_i \in S, \ 1 \le i \le n\right).$ In
combinatorics Combinatorics is an area of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geom ...
, a finite set with elements is sometimes called an ''-set'' and a subset with elements is called a ''-subset''. For example, the set is a 3-set – a finite set with three elements – and is a 2-subset of it. (Those familiar with the definition of the natural numbers themselves as conventional in set theory, the so-called von Neumann construction, may prefer to use the existence of the bijection $f \colon S \to n$, which is equivalent.)

# Basic properties

Any proper
subset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ... of a finite set ''S'' is finite and has fewer elements than ''S'' itself. As a consequence, there cannot exist a
bijection In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ... between a finite set ''S'' and a proper subset of ''S''. Any set with this property is called
Dedekind-finiteIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
. Using the standard ZFC axioms for
set theory Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that are relevant to ...
, every Dedekind-finite set is also finite, but this implication cannot be proved in ZF (Zermelo–Fraenkel axioms without the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ... ) alone. The
axiom of countable choice The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom An axiom, postulate or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The ... , a weak version of the axiom of choice, is sufficient to prove this equivalence. Any injective function between two finite sets of the same cardinality is also a
surjective function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ... (a surjection). Similarly, any surjection between two finite sets of the same cardinality is also an injection. The union of two finite sets is finite, with :$, S\cup T, \le , S, + , T, .$ In fact, by the
inclusion–exclusion principle In combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite set, finite Mathematical structure, structures. It is closely relat ...
: :$, S\cup T, = , S, + , T, - , S\cap T, .$ More generally, the union of any finite number of finite sets is finite. The
Cartesian product In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of finite sets is also finite, with: :$, S\times T, = , S, \times, T, .$ Similarly, the Cartesian product of finitely many finite sets is finite. A finite set with ''n'' elements has 2 distinct subsets. That is, the
power set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
of a finite set is finite, with cardinality 2. Any subset of a finite set is finite. The set of values of a function when applied to elements of a finite set is finite. All finite sets are
countable In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, but not all countable sets are finite. (Some authors, however, use "countable" to mean "countably infinite", so do not consider finite sets to be countable.) The
free semilattice In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
over a finite set is the set of its non-empty subsets, with the
join operation being given by set union.

# Necessary and sufficient conditions for finiteness

In
Zermelo–Fraenkel set theory In set theory illustrating the intersection of two sets Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set ...
without the axiom of choice (ZF), the following conditions are all equivalent: # ''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. # (
Kazimierz Kuratowski Kazimierz Kuratowski (; 2 February 1896 – 18 June 1980) was a Polish Polish may refer to: * Anything from or related to Poland Poland ( pl, Polska ), officially the Republic of Poland ( pl, Rzeczpospolita Polska, links=no ), is a cou ... ) ''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 Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor *Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname *Fred Below (1926–1988), American blues drummer *Fritz von Below (1853� ...
for the set-theoretical formulation of Kuratowski finiteness.) # ( Paul Stäckel) ''S'' can be given a total ordering which is
well-order In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
ed both forwards and backwards. That is, every non-empty subset of ''S'' has both a least and a greatest element in the subset. # Every one-to-one function from ''P''(''P''(''S'')) into itself is
onto In , a surjective function (also known as surjection, or onto function) is a that maps an element to every element ; that is, for every , there is an such that . In other words, every element of the function's is the of one element of its ... . That is, the
powerset Image:Hasse diagram of powerset of 3.svg, 250px, The elements of the power set of order theory, ordered with respect to Inclusion (set theory), inclusion. In mathematics, the power set (or powerset) of a Set (mathematics), set is the set of al ... of the powerset of ''S'' is Dedekind-finite (see below). # Every surjective function from ''P''(''P''(''S'')) onto itself is one-to-one. # (
Alfred Tarski Alfred Tarski (; January 14, 1901 – October 26, 1983), born Alfred Teitelbaum,School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. was a Polish-American logician ...
) Every non-empty family of subsets of ''S'' has a
minimal element In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
with respect to inclusion. (Equivalently, every non-empty family of subsets of ''S'' has a
maximal element In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
with respect to inclusion.) # ''S'' can be well-ordered and any two well-orderings on it are
order isomorphicIn the mathematical field of order theory Order theory is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (g ...
. In other words, the well-orderings on ''S'' have exactly one
order type In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. If the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ... is also assumed (the
axiom of countable choice The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom An axiom, postulate or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The ... is sufficient), then the following conditions are all equivalent: # ''S'' is a finite set. # (
Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to abstract algebra (particularly ring theory In algebra, ring theory is the study of ring (mathematics), rings ...
) Every one-to-one function from ''S'' into itself is onto. # Every surjective function from ''S'' onto itself is one-to-one. # ''S'' is empty or every
partial order In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
ing of ''S'' contains a
maximal element In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
.

# Foundational issues

Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He created set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence be ...
initiated his theory of sets in order to provide a mathematical treatment of infinite sets. Thus the distinction between the finite and the infinite lies at the core of set theory. Certain foundationalists, the strict finitists, reject the existence of infinite sets and thus recommend a mathematics based solely on finite sets. Mainstream mathematicians consider strict finitism too confining, but acknowledge its relative consistency: the universe of
hereditarily finite setIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
s constitutes a model of
Zermelo–Fraenkel set theory In set theory illustrating the intersection of two sets Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set ...
with the
axiom of infinity In axiomatic set theory and the branches of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, cha ...
replaced by its negation. 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. The difficulty stems from
Gödel's incompleteness theorems Gödel's incompleteness theorems are two theorem In mathematics, a theorem is a statement (logic), statement that has been Mathematical proof, proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference r ...
. One can interpret the theory of hereditarily finite sets within
Peano arithmetic In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian people, Italian mathematician Giuseppe Peano. These axioms have been ...
(and certainly also vice versa), so the incompleteness of the theory of Peano arithmetic implies that of the theory of hereditarily finite sets. In particular, there exists a plethora of so-called non-standard models of both theories. 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. (This can happen when the model lacks the sets or functions necessary to witness the infinitude of these sets.) 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. So, at least from the point of view of first-order logic, one can only hope to describe finiteness approximately. More generally, informal notions like set, and particularly finite set, may receive interpretations across a range of
formal system A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A for ...
s varying in their axiomatics and logical apparatus. 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 In the foundations of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theorie ...
(NBG),
Non-well-founded set theoryNon-well-founded set theories are variants of axiomatic set theory illustrating the intersection of two sets. Set theory is a branch of mathematical logic Mathematical logic, also called formal logic, is a subfield of mathematics Mathema ...
,
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell (18 May 1872 – 2 February 1970) was a British polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, "universal human") is an individual whose know ...
's
Type theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
and all the theories of their various models. One may also choose among classical first-order logic, various higher-order logics and
intuitionistic logic Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems of ...
. A formalist might see the meaning of ''set'' varying from system to system. Some kinds of
Platonist Platonism is the philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, Ethics, values, Philosophy of mind, mind, and Philosophy of ...
s might view particular formal systems as approximating an underlying reality.

# Set-theoretic definitions of finiteness

In contexts where the notion of
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
sits logically prior to any notion of set, one can define a set ''S'' as finite if ''S'' admits a
bijection In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ... to some set of
natural numbers In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ... of the form $\$. Mathematicians more typically choose to ground notions of number in
set theory Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that are relevant to ...
, for example they might model natural numbers by the order types of finite
well-ordered In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
sets. Such an approach requires a structural definition of finiteness that does not depend on natural numbers. 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. Two definitions feature prominently in the literature, one due to
Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to abstract algebra (particularly ring theory In algebra, ring theory is the study of ring (mathematics), rings ...
, the other to
Kazimierz Kuratowski Kazimierz Kuratowski (; 2 February 1896 – 18 June 1980) was a Polish Polish may refer to: * Anything from or related to Poland Poland ( pl, Polska ), officially the Republic of Poland ( pl, Rzeczpospolita Polska, links=no ), is a cou ... . (Kuratowski's is the definition used above.) A set ''S'' is called Dedekind infinite if there exists an injective, non-surjective function $f:S \rightarrow S$. Such a function exhibits a bijection between ''S'' and a proper subset of ''S'', namely the image of ''f''. Given a Dedekind infinite set ''S'', a function ''f'', and an element ''x'' that is not in the image of ''f'', we can form an infinite sequence of distinct elements of ''S'', namely $x,f\left(x\right),f\left(f\left(x\right)\right),...$. Conversely, given a sequence in ''S'' consisting of distinct elements $x_1, x_2, x_3, ...$, we can define a function ''f'' such that on elements in the sequence $f\left(x_i\right) = x_$ and ''f'' behaves like the identity function otherwise. Thus Dedekind infinite sets contain subsets that correspond bijectively with the natural numbers. Dedekind finite naturally means that every injective self-map is also surjective. Kuratowski finiteness is defined as follows. Given any set ''S'', the binary operation of union endows the
powerset Image:Hasse diagram of powerset of 3.svg, 250px, The elements of the power set of order theory, ordered with respect to Inclusion (set theory), inclusion. In mathematics, the power set (or powerset) of a Set (mathematics), set is the set of al ... ''P''(''S'') with the structure of a
semilatticeIn mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (mathematics), join (a least upper bound) for any nonempty set, nonempty finite set, finite subset. Duality (order theory), Dually, a meet-semilattic ...
. Writing ''K''(''S'') for the sub-semilattice generated by the
empty set #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ... and the singletons, call set ''S'' Kuratowski finite if ''S'' itself belongs to ''K''(''S''). 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. Readers unfamiliar with semilattices and other notions of abstract algebra may prefer an entirely elementary formulation. Kuratowski finite means ''S'' lies in the set ''K''(''S''), constructed as follows. Write ''M'' for the set of all subsets ''X'' of ''P''(''S'') such that: * ''X'' contains the empty set; * For every set ''T'' in ''P''(''S''), if ''X'' contains ''T'' then ''X'' also contains the union of ''T'' with any singleton. Then ''K''(''S'') may be defined as the intersection of ''M''. In ZF, Kuratowski finite implies Dedekind finite, but not vice versa. 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. 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. However, Kuratowski finiteness would fail for the same set of socks.

## Other concepts of finiteness

In ZF set theory without the axiom of choice, the following concepts of finiteness for a set ''S'' are distinct. 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. 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. (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.) * 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.) * Ia-finite. For every partition of ''S'' into two sets, at least one of the two sets is I-finite. * II-finite. Every non-empty ⊆-monotone set of subsets of ''S'' has a ⊆-maximal element. * III-finite. The power set ''P''(''S'') is Dedekind finite. * IV-finite. ''S'' is Dedekind finite. * V-finite. ∣''S''∣ = 0 or 2⋅∣''S''∣ > ∣''S'', . * VI-finite. ∣''S''∣ = 0 or ∣''S''∣ = 1 or ∣''S''∣2 > ∣''S''∣. * VII-finite. ''S'' 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 In set theory Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that ar ...
s are found using
model theory In mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical ...
. 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. Most of these finiteness definitions and their names are attributed to by . However, definitions I, II, III, IV and V were presented in , 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.

*
FinSet In the mathematical field of category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects ...
*
Ordinal number In set theory Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that ...
*
Peano arithmetic In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian people, Italian mathematician Giuseppe Peano. These axioms have been ...

# References

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