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Zermelo set theory (sometimes denoted by Z-), as set out in a seminal paper in 1908 by
Ernst Zermelo Ernst Friedrich Ferdinand Zermelo (, ; 27 July 187121 May 1953) was a German logician and mathematician, whose work has major implications for the foundations of mathematics. He is known for his role in developing Zermelo–Fraenkel axiomatic se ...
, is the ancestor of modern
Zermelo–Fraenkel set theory In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such a ...
(ZF) and its extensions, such as
von Neumann–Bernays–Gödel set theory In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel–choice set theory (ZFC). NBG introduces the notion of class, which is a colle ...
(NBG). It bears certain differences from its descendants, which are not always understood, and are frequently misquoted. This article sets out the original
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 word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
s, with the original text (translated into English) and original numbering.

# The axioms of Zermelo set theory

The
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 word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
s of Zermelo set theory are stated for objects, some of which (but not necessarily all) are sets, and the remaining objects are
urelements In set theory, a branch of mathematics, an urelement or ur-element (from the German prefix ''ur-'', 'primordial') is an object that is not a set, but that may be an element of a set. It is also referred to as an atom or individual. Theory Th ...
and not sets. Zermelo's language implicitly includes a membership relation ∈, an equality relation = (if it is not included in the underlying logic), and a unary predicate saying whether an object is a set. Later versions of set theory often assume that all objects are sets so there are no urelements and there is no need for the unary predicate. # AXIOM I.
Axiom of extensionality In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo–Fraenkel set theory. It says that sets having the same element ...
(''Axiom der Bestimmtheit'') "If every element of a set ''M'' is also an element of ''N'' and vice versa ... then ''M'' $\equiv$ ''N''. Briefly, every set is determined by its elements." # AXIOM II. Axiom of elementary sets (''Axiom der Elementarmengen'') "There exists a set, the null set, ∅, that contains no element at all. If ''a'' is any object of the domain, there exists a set containing ''a'' and only ''a'' as an element. If ''a'' and ''b'' are any two objects of the domain, there always exists a set containing as elements ''a'' and ''b'' but no object ''x'' distinct from them both." See Axiom of pairs. # AXIOM III.
Axiom of separation In many popular versions of axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation, subset axiom scheme or axiom schema of restricted comprehension is an axiom schema. Essentially, it says that any ...
(''Axiom der Aussonderung'') "Whenever the propositional function –(''x'') is defined for all elements of a set ''M'', ''M'' possesses a subset ''M' '' containing as elements precisely those elements ''x'' of ''M'' for which –(''x'') is true." # AXIOM IV. Axiom of the power set (''Axiom der Potenzmenge'') "To every set ''T'' there corresponds a set ''T' '', the
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is post ...
of ''T'', that contains as elements precisely all subsets of ''T'' ." # AXIOM V. Axiom of the union (''Axiom der Vereinigung'') "To every set ''T'' there corresponds a set ''∪T'', the union of ''T'', that contains as elements precisely all elements of the elements of ''T'' ." # AXIOM VI.
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 of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
(''Axiom der Auswahl'') "If ''T'' is a set whose elements all are sets that are different from ∅ and mutually disjoint, its union ''∪T'' includes at least one subset ''S''1 having one and only one element in common with each element of ''T'' ." # AXIOM VII.
Axiom of infinity In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing th ...
(''Axiom des Unendlichen'') "There exists in the domain at least one set ''Z'' that contains the null set as an element and is so constituted that to each of its elements ''a'' there corresponds a further element of the form , in other words, that with each of its elements ''a'' it also contains the corresponding set as element."

# Connection with standard set theory

The most widely used and accepted set theory is known as ZFC, which consists of
Zermelo–Fraenkel set theory In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such a ...
including 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 of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
(AC). The links show where the axioms of Zermelo's theory correspond. There is no exact match for "elementary sets". (It was later shown that the singleton set could be derived from what is now called the "Axiom of pairs". If ''a'' exists, ''a'' and ''a'' exist, thus exists, and so by extensionality = .) The empty set axiom is already assumed by axiom of infinity, and is now included as part of it. Zermelo set theory does not include the axioms of replacement and regularity. The axiom of replacement was first published in 1922 by
Abraham Fraenkel Abraham Fraenkel ( he, אברהם הלוי (אדולף) פרנקל; February 17, 1891 – October 15, 1965) was a German-born Israeli mathematician. He was an early Zionist and the first Dean of Mathematics at the Hebrew University of Jerusalem. ...
and
Thoralf Skolem Thoralf Albert Skolem (; 23 May 1887 – 23 March 1963) was a Norwegian mathematician who worked in mathematical logic and set theory. Life Although Skolem's father was a primary school teacher, most of his extended family were farmers. Skol ...
, who had independently discovered that Zermelo's axioms cannot prove the existence of the set where ''Z''0 is the set of
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal ...
s and ''Z''''n''+1 is the
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is post ...
of ''Z''''n''. They both realized that the axiom of replacement is needed to prove this. The following year,
John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...
pointed out that the axiom of regularity is necessary to build his theory of ordinals. The axiom of regularity was stated by von Neumann in 1925. In the modern ZFC system, the "propositional function" referred to in the axiom of separation is interpreted as "any property definable by a first-order
formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a '' chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betw ...
with parameters", so the separation axiom is replaced by an
axiom schema In mathematical logic, an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion of axiom. Formal definition An axiom schema is a formula in the metalanguage of an axiomatic system, in which one or more schematic variables ...
. The notion of "first order formula" was not known in 1908 when Zermelo published his axiom system, and he later rejected this interpretation as being too restrictive. Zermelo set theory is usually taken to be a first-order theory with the separation axiom replaced by an axiom scheme with an axiom for each first-order formula. It can also be considered as a theory in
second-order logic In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory. First-order logic quantifies o ...
, where now the separation axiom is just a single axiom. The second-order interpretation of Zermelo set theory is probably closer to Zermelo's own conception of it, and is stronger than the first-order interpretation. In the usual cumulative hierarchy ''V''α of ZFC set theory (for ordinals α), any one of the sets ''V''α for α a limit ordinal larger than the first infinite ordinal ω (such as ''V''ω·2) forms a model of Zermelo set theory. So the consistency of Zermelo set theory is a theorem of ZFC set theory. As $V_$ models Zermelo's axioms while not containing $\aleph_\omega$ and larger infinite cardinals, by
Gödel's completeness theorem Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. The completeness theorem applies to any first-order theory: I ...
Zermelo's axioms do not prove the existence of these cardinals. (Cardinals have to be defined differently in Zermelo set theory, as the usual definition of cardinals and ordinals does not work very well: with the usual definition it is not even possible to prove the existence of the ordinal ω2.) The
axiom of infinity In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing th ...
is usually now modified to assert the existence of the first infinite von Neumann ordinal $\omega$; the original Zermelo axioms cannot prove the existence of this set, nor can the modified Zermelo axioms prove Zermelo's axiom of infinity. Zermelo's axioms (original or modified) cannot prove the existence of $V_$ as a set nor of any rank of the cumulative hierarchy of sets with infinite index. Zermelo allowed for the existence of
urelements In set theory, a branch of mathematics, an urelement or ur-element (from the German prefix ''ur-'', 'primordial') is an object that is not a set, but that may be an element of a set. It is also referred to as an atom or individual. Theory Th ...
that are not sets and contain no elements; these are now usually omitted from set theories.

# Mac Lane set theory

Mac Lane set theory, introduced by , is Zermelo set theory with the axiom of separation restricted to first-order formulas in which every quantifier is bounded. Mac Lane set theory is similar in strength to
topos theory In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notion ...
with a
natural number object In category theory, a natural numbers object (NNO) is an object endowed with a recursive structure similar to natural numbers. More precisely, in a category E with a terminal object 1, an NNO ''N'' is given by: # a global element ''z'' : 1 → '' ...
, or to the system in
Principia mathematica The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. ...
. It is strong enough to carry out almost all ordinary mathematics not directly connected with set theory or logic.

# The aim of Zermelo's paper

The introduction states that the very existence of the discipline of set theory "seems to be threatened by certain contradictions or "antinomies", that can be derived from its principles – principles necessarily governing our thinking, it seems – and to which no entirely satisfactory solution has yet been found". Zermelo is of course referring to the " Russell antinomy". He says he wants to show how the original theory of
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance o ...
and
Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. His ...
can be reduced to a few definitions and seven principles or axioms. He says he has ''not'' been able to prove that the axioms are consistent. A non-constructivist argument for their consistency goes as follows. Define ''V''α for α one of the ordinals 0, 1, 2, ...,ω, ω+1, ω+2,..., ω·2 as follows: * V0 is the empty set. * For α a successor of the form β+1, ''V''α is defined to be the collection of all subsets of ''V''β. * For α a limit (e.g. ω, ω·2) then ''V''α is defined to be the union of ''V''β for β<α. Then the axioms of Zermelo set theory are consistent because they are true in the model ''V''ω·2. While a non-constructivist might regard this as a valid argument, a constructivist would probably not: while there are no problems with the construction of the sets up to ''V''ω, the construction of ''V''ω+1 is less clear because one cannot constructively define every subset of ''V''ω. This argument can be turned into a valid proof with the addition of a single new axiom of infinity to Zermelo set theory, simply that ''V''ω·2 ''exists''. This is presumably not convincing for a constructivist, but it shows that the consistency of Zermelo set theory can be proved with a theory which is not very different from Zermelo theory itself, only a little more powerful.

# The axiom of separation

Zermelo comments that Axiom III of his system is the one responsible for eliminating the antinomies. It differs from the original definition by Cantor, as follows. Sets cannot be independently defined by any arbitrary logically definable notion. They must be constructed in some way from previously constructed sets. For example, they can be constructed by taking powersets, or they can be ''separated'' as subsets of sets already "given". This, he says, eliminates contradictory ideas like "the set of all sets" or "the set of all ordinal numbers". He disposes of the
Russell paradox In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains a ...
by means of this Theorem: "Every set $M$ possesses at least one subset $M_0$ that is not an element of $M$ ". Let $M_0$ be the subset of $M$ for which, by AXIOM III, is separated out by the notion "$x \notin x$". Then $M_0$ cannot be in $M$. For # If $M_0$ is in $M_0$, then $M_0$ contains an element ''x'' for which ''x'' is in ''x'' (i.e. $M_0$ itself), which would contradict the definition of $M_0$. # If $M_0$ is not in $M_0$, and assuming $M_0$ is an element of ''M'', then $M_0$ is an element of ''M'' that satisfies the definition "$x \notin x$", and so is in $M_0$ which is a contradiction. Therefore, the assumption that $M_0$ is in $M$ is wrong, proving the theorem. Hence not all objects of the universal domain ''B'' can be elements of one and the same set. "This disposes of the Russell
antinomy Antinomy ( Greek ἀντί, ''antí'', "against, in opposition to", and νόμος, ''nómos'', "law") refers to a real or apparent mutual incompatibility of two laws. It is a term used in logic and epistemology, particularly in the philosophy of ...
as far as we are concerned". This left the problem of "the domain ''B''" which seems to refer to something. This led to the idea of a
proper class Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map for ...
.

# Cantor's theorem

Zermelo's paper may be the first to mention the name "
Cantor's theorem In mathematical set theory, Cantor's theorem is a fundamental result which states that, for any set A, the set of all subsets of A, the power set of A, has a strictly greater cardinality than A itself. For finite sets, Cantor's theorem can ...
". Cantor's theorem: "If ''M'' is an arbitrary set, then always ''M'' < P(''M'') he power set of ''M'' Every set is of lower cardinality than the set of its subsets". Zermelo proves this by considering a function φ: ''M'' → P(''M''). By Axiom III this defines the following set ''M' '': :''M' '' = . But no element ''m' '' of ''M '' could correspond to ''M' '', i.e. such that φ(''m' '') = ''M' ''. Otherwise we can construct a contradiction: :1) If ''m' '' is in ''M' '' then by definition ''m' '' ∉ φ(''m' '') = ''M' '', which is the first part of the contradiction :2) If ''m' '' is not in ''M' '' but in ''M '' then by definition ''m' '' ∉ ''M' '' = φ(''m' '') which by definition implies that ''m' '' is in ''M' '', which is the second part of the contradiction. so by contradiction ''m' '' does not exist. Note the close resemblance of this proof to the way Zermelo disposes of Russell's paradox.