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set theory 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, set theory, as a branch of mathematics, is mostly concern ...
, the axiom of limitation of size was proposed by
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 c ...
in his 1925
axiom system In mathematics and logic, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contain ...
for
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
s and
class Class or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used differently ...
es.; English translation: . It formalizes the limitation of size principle, which avoids the paradoxes encountered in earlier formulations of
set theory 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, set theory, as a branch of mathematics, is mostly concern ...
by recognizing that some classes are too big to be sets. Von Neumann realized that the paradoxes are caused by permitting these big classes to be members of a class.. A class that is a member of a class is a set; a class that is not a set is 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 f ...
. Every class is a subclass of '' V'', the class of all sets. The axiom of limitation of size says that a class is a set if and only if it is smaller than ''V''—that is, there is no function mapping it onto ''V''. Usually, this axiom is stated in the equivalent form: A class is a proper class if and only if there is a function that maps it onto ''V''. Von Neumann's axiom implies the axioms of replacement, separation, union, and global choice. It is equivalent to the combination of replacement, union, and global choice in
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) and
Morse–Kelley set theory In the foundations of mathematics, Morse–Kelley set theory (MK), Kelley–Morse set theory (KM), Morse–Tarski set theory (MT), Quine–Morse set theory (QM) or the system of Quine and Morse is a first-order axiomatic set theory that is closely ...
. Later expositions of class theories—such as those of
Paul Bernays Paul Isaac Bernays (17 October 1888 – 18 September 1977) was a Swiss mathematician who made significant contributions to mathematical logic, axiomatic set theory, and the philosophy of mathematics. He was an assistant and close collaborator of ...
,
Kurt Gödel Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an imm ...
, and John L. Kelley—use replacement, union, and a choice axiom equivalent to global choice rather than von Neumann's axiom. In 1930,
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 ...
defined models of set theory satisfying the axiom of limitation of size.; English translation: . Abraham Fraenkel and
Azriel Lévy Azriel Lévy (Hebrew: עזריאל לוי; born c. 1934) is an Israeli mathematician, logician, and a professor emeritus at the Hebrew University of Jerusalem. Biography Lévy obtained his Ph.D. at the Hebrew University of Jerusalem in 1958, un ...
have stated that the axiom of limitation of size does not capture all of the "limitation of size doctrine" because it does not imply the power set axiom. Michael Hallett has argued that the limitation of size doctrine does not justify the power set axiom and that "von Neumann's explicit assumption f the smallness of power-setsseems preferable to Zermelo's, Fraenkel's, and Lévy's obscurely hidden ''implicit'' assumption of the smallness of power-sets.".


Formal statement

The usual version of the axiom of limitation of size—a class is a proper class if and only if there is a function that maps it onto ''V''—is expressed in the
formal language In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules. The alphabet of a formal language consists of sym ...
of set theory as: :\begin \forall C \Bigl \lnot \exist D \left(C \in D\right) \iff \exist F \bigl[&\,\forall y \bigl(\exist D(y \in D) \implies \exist x [\,x \in C \land (x, y) \in F\,bigr) \\ &\, \land \, \forall x \forall y \forall z \bigl(\,[\,(x, y) \in F \land (x, z) \in F\,] \implies y = z\bigr)\,\bigr]\,\Bigr] \end Gödel introduced the convention that uppercase variables range over all the classes, while lowercase variables range over all the sets. This convention allows us to write: * \exist y\, \varphi(y) instead of \exist y \bigl(\exist D (y \in D) \land \varphi(y)\bigr) * \forall y\, \varphi(y) instead of \forall y \bigl(\exist D (y \in D) \implies \varphi(y)\bigr) With Gödel's convention, the axiom of limitation of size can be written: :\begin \forall C \Bigl[ \lnot \exist D \left( C \in D\right) \iff \exist F \bigl[&\,\forall y \exist x \bigl( x \in C \land (x, y) \in F \bigr) \\ &\, \land \, \forall x \forall y \forall z \bigl(\,[\,(x, y) \in F \land (x, z) \in F\,] \implies y = z\bigr)\,\bigr]\,\Bigr] \end


Implications of the axiom

Von Neumann proved that the axiom of limitation of size implies the axiom of replacement, which can be expressed as: If ''F'' is a function and ''A'' is a set, then ''F''(''A'') is a set. This is proved by contradiction. Let ''F'' be a function and ''A'' be a set. Assume that ''F''(''A'') is a proper class. Then there is a function ''G'' that maps ''F''(''A'') onto ''V''. Since the
composite function In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
''G'' ∘ ''F'' maps ''A'' onto ''V'', the axiom of limitation of size implies that ''A'' is a proper class, which contradicts ''A'' being a set. Therefore, ''F''(''A'') is a set. Since the axiom of replacement implies the axiom of separation, the axiom of limitation of size implies the
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 ...
. Von Neumann also proved that his axiom implies that ''V'' can be
well-ordered In mathematics, a well-order (or well-ordering or well-order relation) on a set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together with the well-or ...
. The proof starts by proving by contradiction that ''Ord'', the class of all ordinals, is a proper class. Assume that ''Ord'' is a set. Since it is a
transitive set In set theory, a branch of mathematics, a set A is called transitive if either of the following equivalent conditions hold: * whenever x \in A, and y \in x, then y \in A. * whenever x \in A, and x is not an urelement, then x is a subset of A. Simil ...
that is strictly well-ordered by ∈, it is an ordinal. So ''Ord'' ∈ ''Ord'', which contradicts ''Ord'' being strictly well-ordered by ∈. Therefore, ''Ord'' is a proper class. So von Neumann's axiom implies that there is a function ''F'' that maps ''Ord'' onto ''V''. To define a well-ordering of ''V'', let ''G'' be the subclass of ''F'' consisting of the ordered pairs (α, ''x'') where α is the least β such that (β, ''x'') ∈ ''F''; that is, ''G'' = . The function ''G'' is a
one-to-one correspondence In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
between a subset of ''Ord'' and ''V''. Therefore, ''x'' < ''y'' if ''G''−1(x) < ''G''−1(y) defines a well-ordering of ''V''. This well-ordering defines a global
choice function A choice function (selector, selection) is a mathematical function ''f'' that is defined on some collection ''X'' of nonempty sets and assigns some element of each set ''S'' in that collection to ''S'' by ''f''(''S''); ''f''(''S'') maps ''S'' to ...
: Let ''Inf''(''x'') be the least element of a non-empty set ''x''. Since ''Inf''(''x'') ∈ ''x'', this function chooses an element of ''x'' for every non-empty set ''x''. Therefore, ''Inf''(''x'') is a global choice function, so Von Neumann's axiom implies the
axiom of global choice In mathematics, specifically in class theories, the axiom of global choice is a stronger variant of the axiom of choice that applies to proper classes of sets as well as sets of sets. Informally it states that one can simultaneously choose an ele ...
. In 1968,
Azriel Lévy Azriel Lévy (Hebrew: עזריאל לוי; born c. 1934) is an Israeli mathematician, logician, and a professor emeritus at the Hebrew University of Jerusalem. Biography Lévy obtained his Ph.D. at the Hebrew University of Jerusalem in 1958, un ...
proved that von Neumann's axiom implies the axiom of union. First, he proved without using the axiom of union that every set of ordinals has an upper bound. Then he used a function that maps ''Ord'' onto ''V'' to prove that if ''A'' is a set, then ∪A is a set. The axioms of replacement, global choice, and union (with the other axioms of NBG) imply the axiom of limitation of size. Therefore, this axiom is equivalent to the combination of replacement, global choice, and union in NBG or
Morse–Kelley set theory In the foundations of mathematics, Morse–Kelley set theory (MK), Kelley–Morse set theory (KM), Morse–Tarski set theory (MT), Quine–Morse set theory (QM) or the system of Quine and Morse is a first-order axiomatic set theory that is closely ...
. These set theories only substituted the axiom of replacement and a form of the axiom of choice for the axiom of limitation of size because von Neumann's axiom system contains the axiom of union. Lévy's proof that this axiom is redundant came many years later. The axioms of NBG with the axiom of global choice replaced by the usual
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 ...
do not imply the axiom of limitation of size. In 1964, William B. Easton used forcing to build a
model A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. Models c ...
of NBG with global choice replaced by the axiom of choice. In Easton's model, ''V'' cannot be linearly ordered, so it cannot be well-ordered. Therefore, the axiom of limitation of size fails in this model. ''Ord'' is an example of a proper class that cannot be mapped onto ''V'' because (as proved above) if there is a function mapping ''Ord'' onto ''V'', then ''V'' can be well-ordered. The axioms of NBG with the axiom of replacement replaced by the weaker axiom of separation do not imply the axiom of limitation of size. Define \omega_\alpha as the \alpha-th infinite initial ordinal, which is also the
cardinal Cardinal or The Cardinal may refer to: Animals * Cardinal (bird) or Cardinalidae, a family of North and South American birds **'' Cardinalis'', genus of cardinal in the family Cardinalidae **'' Cardinalis cardinalis'', or northern cardinal, t ...
\aleph_\alpha; numbering starts at 0, so \omega_0 = \omega. In 1939, Gödel pointed out that Lωω, a subset of the constructible universe, is a model of ZFC with replacement replaced by separation. To expand it into a model of NBG with replacement replaced by separation, let its classes be the sets of Lωω+1, which are the constructible subsets of Lωω. This model satisfies NBG's class existence axioms because restricting the set variables of these axioms to Lωω produces instances of the axiom of separation, which holds in L. It satisfies the axiom of global choice because there is a function belonging to Lωω+1 that maps ωω onto Lωω, which implies that Lωω is well-ordered. The axiom of limitation of size fails because the proper class has cardinality \aleph_0, so it cannot be mapped onto Lωω, which has cardinality \aleph_\omega. In a 1923 letter to Zermelo, von Neumann stated the first version of his axiom: A class is a proper class if and only if there is a one-to-one correspondence between it and ''V''. The axiom of limitation of size implies von Neumann's 1923 axiom. Therefore, it also implies that all proper classes are
equinumerous In mathematics, two sets or classes ''A'' and ''B'' are equinumerous if there exists a one-to-one correspondence (or bijection) between them, that is, if there exists a function from ''A'' to ''B'' such that for every element ''y'' of ''B'', the ...
with ''V''.


Zermelo's models and the axiom of limitation of size

In 1930, Zermelo published an article on models of set theory, in which he proved that some of his models satisfy the axiom of limitation of size. These models are built in ZFC by using the
cumulative hierarchy In mathematics, specifically set theory, a cumulative hierarchy is a family of sets W_\alpha indexed by ordinals \alpha such that * W_\alpha \subseteq W_ * If \lambda is a limit ordinal, then W_\lambda = \bigcup_ W_ Some authors additionally r ...
''V''α, which is defined by
transfinite recursion Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC. Induction by cases Let P(\alpha) be a property defined for a ...
: # ''V''0 = 
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
. # ''V''α+1 = ''V''α ∪ ''P''(''V''α). That is, the union of ''V''α and its
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 ...
. # For limit β: ''V''β = ∪α < β ''V''α. That is, ''V''β is the union of the preceding ''V''α. Zermelo worked with models of the form ''V''κ where κ is a
cardinal Cardinal or The Cardinal may refer to: Animals * Cardinal (bird) or Cardinalidae, a family of North and South American birds **'' Cardinalis'', genus of cardinal in the family Cardinalidae **'' Cardinalis cardinalis'', or northern cardinal, t ...
. The classes of the model are the
subsets In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
of ''V''κ, and the model's ∈-relation is the standard ∈-relation. The sets of the model are the classes ''X'' such that ''X'' ∈ ''V''κ. Zermelo identified cardinals κ such that ''V''κ satisfies: : Theorem 1. A class ''X'' is a set if and only if , ''X'',  < κ. : Theorem 2. , ''V''κ,  = κ. Since every class is a subset of ''V''κ, Theorem 2 implies that every class ''X'' has
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
 ≤ κ. Combining this with Theorem 1 proves: every proper class has cardinality κ. Hence, every proper class can be put into one-to-one correspondence with ''V''κ. This correspondence is a subset of ''V''κ, so it is a class of the model. Therefore, the axiom of limitation of size holds for the model ''V''κ. The theorem stating that ''V''κ has a well-ordering can be proved directly. Since κ is an ordinal of cardinality κ and , ''V''κ,  = κ, there is a
one-to-one correspondence In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
between κ and ''V''κ. This correspondence produces a well-ordering of ''V''κ. Von Neumann's proof is indirect. It uses the
Burali-Forti paradox In set theory, a field of mathematics, the Burali-Forti paradox demonstrates that constructing "the set of all ordinal numbers" leads to a contradiction and therefore shows an antinomy in a system that allows its construction. It is named after C ...
to prove by contradiction that the class of all ordinals is a proper class. Hence, the axiom of limitation of size implies that there is a function that maps the class of all ordinals onto the class of all sets. This function produces a well-ordering of ''V''κ.


The model ''V''ω

To demonstrate that Theorems 1 and 2 hold for some ''V''κ, we first prove that if a set belongs to ''V''α then it belongs to all subsequent ''V''β, or equivalently: ''V''α ⊆ ''V''β for α ≤ β. This is proved by
transfinite induction Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC. Induction by cases Let P(\alpha) be a property defined for ...
on β: # β = 0: ''V''0 ⊆ ''V''0. # For β+1: By inductive hypothesis, ''V''α ⊆ ''V''β. Hence, ''V''α ⊆ ''V''β ⊆ ''V''β ∪ ''P''(''V''β) = ''V''β+1. # For limit β: If α < β, then ''V''α ⊆ ∪ξ < β ''V''ξ = ''V''β. If α = β, then ''V''α ⊆ ''V''β. Sets enter the cumulative hierarchy through the power set ''P''(''V''β) at step β+1. The following definitions will be needed: :If ''x'' is a set,
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * ...
(''x'') is the least ordinal β such that ''x'' ∈ ''V''β+1. :The
supremum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest ...
of a set of ordinals A, denoted by sup A, is the least ordinal β such that α ≤ β for all α ∈ A. Zermelo's smallest model is ''V''ω.
Mathematical induction Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ...  all hold. Informal metaphors help ...
proves that ''V''''n'' is
finite Finite is the opposite of 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 or marke ...
for all ''n'' < ω: # , ''V''0,  = 0. # , ''V''''n''+1,  = , ''V''''n'' ∪ ''P''(''V''''n''),  ≤ , ''V''''n'',  + 2 , ''V''''n'', , which is finite since ''V''''n'' is finite by inductive hypothesis. Proof of Theorem 1: A set ''X'' enters ''V''ω through ''P''(''V''''n'') for some ''n'' < ω, so ''X'' ⊆ ''V''''n''. Since ''V''''n'' is finite, ''X'' is finite.
Conversely In logic and mathematics, the converse of a categorical or implicational statement is the result of reversing its two constituent statements. For the implication ''P'' → ''Q'', the converse is ''Q'' → ''P''. For the categorical proposit ...
: If a class ''X'' is finite, let ''N'' = sup . Since rank(''x'') ≤ ''N'' for all ''x'' ∈ ''X'', we have ''X'' ⊆ ''V''''N''+1, so ''X'' ∈ ''V''''N''+2 ⊆ ''V''ω. Therefore, ''X'' ∈ ''V''ω. Proof of Theorem 2: ''V''ω is the union of
countably infinite In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural number ...
ly many finite sets of increasing size. Hence, it has cardinality \aleph_0, which equals ω by von Neumann cardinal assignment. The sets and classes of ''V''ω satisfy all the axioms of NBG except 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 ...
.


The models ''V''κ where κ is a strongly inaccessible cardinal

Two properties of finiteness were used to prove Theorems 1 and 2 for ''V''ω: # If λ is a finite cardinal, then 2λ is finite. # If ''A'' is a set of ordinals such that , ''A'', is finite, and α is finite for all α ∈ ''A'', then sup ''A'' is finite. To find models satisfying the axiom of infinity, replace "finite" by "< κ" to produce the properties that define
strongly inaccessible cardinal In set theory, an uncountable cardinal is inaccessible if it cannot be obtained from smaller cardinals by the usual operations of cardinal arithmetic. More precisely, a cardinal is strongly inaccessible if it is uncountable, it is not a sum of ...
s. A cardinal κ is strongly inaccessible if κ > ω and: # If λ is a cardinal such that λ < κ, then 2λ < κ. # If ''A'' is a set of ordinals such that , ''A'',  < κ, and α < κ for all α ∈ ''A'', then sup ''A'' < κ. These properties assert that κ cannot be reached from below. The first property says κ cannot be reached by power sets; the second says κ cannot be reached by the axiom of replacement. Just as the axiom of infinity is required to obtain ω, an axiom is needed to obtain strongly inaccessible cardinals. Zermelo postulated the existence of an unbounded sequence of strongly inaccessible cardinals. If κ is a strongly inaccessible cardinal, then transfinite induction proves , ''V''α,  < κ for all α < κ: # α = 0: , ''V''0,  = 0. # For α+1: , ''V''α+1,  = , ''V''α ∪ ''P''(''V''α),  ≤ , ''V''α,  + 2 , ''V''α,  = 2 , ''V''α,  < κ. Last inequality uses inductive hypothesis and κ being strongly inaccessible. # For limit α: , ''V''α,  = , ∪ξ < α ''V''ξ,  ≤ sup  < κ. Last inequality uses inductive hypothesis and κ being strongly inaccessible. Proof of Theorem 1: A set ''X'' enters ''V''κ through ''P''(''V''α) for some α < κ, so ''X'' ⊆ ''V''α. Since , ''V''α,  < κ, we obtain , ''X'',  < κ. Conversely: If a class ''X'' has , ''X'',  < κ, let β = sup . Because κ is strongly inaccessible, , ''X'',  < κ and rank(''x'') < κ for all ''x'' ∈ ''X'' imply β = sup  < κ. Since rank(''x'') ≤ β for all ''x'' ∈ ''X'', we have ''X'' ⊆ ''V''β+1, so ''X'' ∈ ''V''β+2 ⊆ ''V''κ. Therefore, ''X'' ∈ ''V''κ. Proof of Theorem 2: , ''V''κ,  = , ∪α < κ ''V''α,  ≤ sup . Let β be this supremum. Since each ordinal in the supremum is less than κ, we have β ≤ κ. Assume β < κ. Then there is a cardinal λ such that β < λ < κ; for example, let λ = 2, β, . Since λ ⊆ ''V''λ and , ''V''λ, is in the supremum, we have λ ≤ , ''V''λ,  ≤ β. This contradicts β < λ. Therefore, , ''V''κ,  = β = κ. The sets and classes of ''V''κ satisfy all the axioms of NBG.


Limitation of size doctrine

The limitation of size doctrine is a
heuristic A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate ...
principle that is used to justify axioms of set theory. It avoids the set theoretical paradoxes by restricting the full (contradictory) comprehension axiom schema: :\forall w_1,\ldots,w_n \, \exists x \, \forall u \, ( u \in x \iff \varphi(u, w_1, \ldots, w_n) ) to instances "that do not give sets 'too much bigger' than the ones they use." If "bigger" means "bigger in cardinal size," then most of the axioms can be justified: The axiom of separation produces a subset of ''x'' that is not bigger than ''x''. The axiom of replacement produces an image set ''f''(''x'') that is not bigger than ''x''. The axiom of union produces a union whose size is not bigger than the size of the biggest set in the union times the number of sets in the union. The axiom of choice produces a choice set whose size is not bigger than the size of the given set of nonempty sets. The limitation of size doctrine does not justify the axiom of infinity: :\exists y \, empty \in y \, \land \, \forall x (x \in y \implies x \cup \ \in y) which uses the
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in othe ...
and sets obtained from the empty set by iterating the ordinal successor operation. Since these sets are finite, any set satisfying this axiom, such as ω, is much bigger than these sets. Fraenkel and Lévy regard the empty set and the infinite 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 ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
, whose existence is implied by the axioms of infinity and separation, as the starting point for generating sets. Von Neumann's approach to limitation of size uses the axiom of limitation of size. As mentioned in , von Neumann's axiom implies the axioms of separation, replacement, union, and choice. Like Fraenkel and Lévy, von Neumann had to add the axiom of infinity to his system since it cannot be proved from his other axioms. The differences between von Neumann's approach to limitation of size and Fraenkel and Lévy's approach are: * Von Neumann's axiom puts limitation of size into an axiom system, making it possible to prove most set existence axioms. The limitation of size doctrine justifies axioms using informal arguments that are more open to disagreement than a proof. * Von Neumann assumed the power set axiom since it cannot be proved from his other axioms. Fraenkel and Lévy state that the limitation of size doctrine justifies the power set axiom. There is disagreement on whether the limitation of size doctrine justifies the power set axiom. Michael Hallett has analyzed the arguments given by Fraenkel and Lévy. Some of their arguments measure size by criteria other than cardinal size—for example, Fraenkel introduces "comprehensiveness" and "extendability." Hallett points out what he considers to be flaws in their arguments. Hallett then argues that results in set theory seem to imply that there is no link between the size of an infinite set and the size of its power set. This would imply that the limitation of size doctrine is incapable of justifying the power set axiom because it requires that the power set of ''x'' is not "too much bigger" than ''x''. For the case where size is measured by cardinal size, Hallett mentions
Paul Cohen Paul Joseph Cohen (April 2, 1934 – March 23, 2007) was an American mathematician. He is best known for his proofs that the continuum hypothesis and the axiom of choice are independent from Zermelo–Fraenkel set theory, for which he was award ...
's work.. Starting with a model of ZFC and \aleph_\alpha, Cohen built a model in which the cardinality of the power set of ω is \aleph_\alpha if the
cofinality In mathematics, especially in order theory, the cofinality cf(''A'') of a partially ordered set ''A'' is the least of the cardinalities of the cofinal subsets of ''A''. This definition of cofinality relies on the axiom of choice, as it uses t ...
of \aleph_\alpha is not ω; otherwise, its cardinality is \aleph_. Since the cardinality of the power set of ω has no bound, there is no link between the cardinal size of ω and the cardinal size of ''P''(ω). Hallett also discusses the case where size is measured by "comprehensiveness," which considers a collection "too big" if it is of "unbounded comprehension" or "unlimited extent." He points out that for an infinite set, we cannot be sure that we have all its subsets without going through the unlimited extent of the universe. He also quotes
John L. Bell John Lamberton Bell (born 1949) is a Scottish hymn-writer and Church of Scotland minister. He is a member of the Iona Community, a broadcaster, and former student activist. He works throughout the world, lecturing in theological colleges in th ...
and
Moshé Machover Moshé Machover ( he, משה מחובר; born 1936) is a mathematician, philosopher, and socialist activist, noted for his writings against Zionism. Born to a Jewish family in Tel Aviv, then part of the British Mandate of Palestine, Machover move ...
: "... the power set ''P''(''u'') of a given nfiniteset ''u'' is proportional not only to the size of ''u'' but also to the 'richness' of the entire universe ..." After making these observations, Hallett states: "One is led to suspect that there is simply ''no link'' between the size (comprehensiveness) of an infinite ''a'' and the size of ''P''(''a'')." Hallett considers the limitation of size doctrine valuable for justifying most of the axioms of set theory. His arguments only indicate that it cannot justify the axioms of infinity and power set. He concludes that "von Neumann's explicit assumption f the smallness of power-setsseems preferable to Zermelo's, Fraenkel's, and Lévy's obscurely hidden ''implicit'' assumption of the smallness of power-sets."


History

Von Neumann developed the axiom of limitation of size as a new method of identifying sets. ZFC identifies sets via its set building axioms. However, as Abraham Fraenkel pointed out: "The rather arbitrary character of the processes which are chosen in the axioms of Z FCas the basis of the theory, is justified by the historical development of set-theory rather than by logical arguments." The historical development of the ZFC axioms began in 1908 when Zermelo chose axioms to eliminate the paradoxes and to support his proof of the
well-ordering theorem In mathematics, the well-ordering theorem, also known as Zermelo's theorem, states that every set can be well-ordered. A set ''X'' is ''well-ordered'' by a strict total order if every non-empty subset of ''X'' has a least element under the orde ...
. In 1922, Abraham Fraenkel 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. Skolem ...
pointed out 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 of ''Z''''n''. They also introduced the axiom of replacement, which guarantees the existence of this set. However, adding axioms as they are needed neither guarantees the existence of all reasonable sets nor clarifies the difference between sets that are safe to use and collections that lead to contradictions. In a 1923 letter to Zermelo, von Neumann outlined an approach to set theory that identifies sets that are "too big" and might lead to contradictions. Von Neumann identified these sets using the criterion: "A set is 'too big' if and only if it is equivalent with the set of all things." He then restricted how these sets may be used: "... in order to avoid the paradoxes those
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which are 'too big' are declared to be impermissible as ''elements''." By combining this restriction with his criterion, von Neumann obtained his first version of the axiom of limitation of size, which in the language of classes states: A class is a proper class if and only if it is equinumerous with ''V''. By 1925, Von Neumann modified his axiom by changing "it is equinumerous with ''V''" to "it can be mapped onto ''V''", which produces the axiom of limitation of size. This modification allowed von Neumann to give a simple proof of the axiom of replacement. Von Neumann's axiom identifies sets as classes that cannot be mapped onto ''V''. Von Neumann realized that, even with this axiom, his set theory does not fully characterize sets. Gödel found von Neumann's axiom to be "of great interest": :"In particular I believe that his on Neumann'snecessary and sufficient condition which a property must satisfy, in order to define a set, is of great interest, because it clarifies the relationship of axiomatic set theory to the paradoxes. That this condition really gets at the essence of things is seen from the fact that it implies the axiom of choice, which formerly stood quite apart from other existential principles. The inferences, bordering on the paradoxes, which are made possible by this way of looking at things, seem to me, not only very elegant, but also very interesting from the logical point of view. Moreover I believe that only by going farther in this direction, i.e., in the direction opposite to constructivism, will the basic problems of abstract set theory be solved."From a Nov. 8, 1957 letter Gödel wrote to
Stanislaw Ulam Stanisław Marcin Ulam (; 13 April 1909 – 13 May 1984) was a Polish-American scientist in the fields of mathematics and nuclear physics. He participated in the Manhattan Project, originated the Teller–Ulam design of thermonuclear weapon ...
().


Notes


References


Bibliography

* . * . * . * . * . * . * . * . * . * . * . * . * . * . * . * . * . * . *. * . ** English translation: . * . ** English translation: . * . * . ** English translation: . * . ** English translation: . {{Set theory Axioms of set theory