{"history":[],"article":{"id":"925fca26-43f2-4c36-b124-49ef4cf65f4b","title":"Axiom of limitation of size","slug":"axiom-of-limitation-of-size","content":":::summary\n:::\n\n# \nIn [set theory](https://bliptext.com/articles/set-theory), the **axiom of limitation of size** was proposed by [John von Neumann](https://bliptext.com/articles/john-von-neumann) in his 1925 [axiom system](https://bliptext.com/articles/axiom-system) for [sets](https://bliptext.com/articles/set-mathematics) and [classes](https://bliptext.com/articles/class-set-theory).\nIt formalizes the [limitation of size](https://bliptext.com/articles/limitation-of-size) principle, which avoids the paradoxes encountered in earlier formulations of [set theory](https://bliptext.com/articles/set-theory) by recognizing that some classes are too big to be sets.\nVon Neumann realized that the paradoxes are caused by permitting these big classes to be members of a class.\nA class that is a member of a class is a set; a class that is not a set is a [proper class](https://bliptext.com/articles/proper-class).\nEvery class is a [subclass](https://bliptext.com/articles/subclass-set-theory) of [V](https://bliptext.com/articles/von-neumann-universe), the class of all sets.\nThe 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](https://bliptext.com/articles/onto) V.\nUsually, this axiom is stated in the [equivalent](https://bliptext.com/articles/logically-equivalent) form: A class is a proper class if and only if there is a function that maps it onto V.\nVon Neumann's axiom implies the axioms of [replacement](https://bliptext.com/articles/axiom-of-replacement), [separation](https://bliptext.com/articles/axiom-of-separation), [union](https://bliptext.com/articles/axiom-of-union), and [global choice](https://bliptext.com/articles/axiom-of-global-choice).\nIt is equivalent to the combination of replacement, union, and global choice in [Von Neumann–Bernays–Gödel set theory](https://bliptext.com/articles/von-neumann-bernays-g-del-set-theory) (NBG) and [Morse–Kelley set theory](https://bliptext.com/articles/morse-kelley-set-theory).\nLater expositions of class theories—such as those of [Paul Bernays](https://bliptext.com/articles/paul-bernays), [Kurt Gödel](https://bliptext.com/articles/kurt-g-del), and [John L. Kelley](https://bliptext.com/articles/john-l-kelley)—use replacement, union, and a choice axiom equivalent to global choice rather than von Neumann's axiom.\nIn 1930, [Ernst Zermelo](https://bliptext.com/articles/ernst-zermelo) defined models of set theory satisfying the axiom of limitation of size.\n[Abraham Fraenkel](https://bliptext.com/articles/abraham-fraenkel) and [Azriel Lévy](https://bliptext.com/articles/azriel-l-vy) 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](https://bliptext.com/articles/axiom-of-power-set).\n[Michael Hallett](https://bliptext.com/articles/michael-hallett) has argued that the limitation of size doctrine does not justify the power set axiom and that \"von Neumann's explicit assumption [of the smallness of power-sets] seems preferable to Zermelo's, Fraenkel's, and Lévy's obscurely hidden implicit assumption of the smallness of power-sets.\"\n\n# Formal statement\nThe 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](https://bliptext.com/articles/formal-language) of set theory as:\nGödel introduced the convention that uppercase variables range over all the classes, while lowercase variables range over all the sets.\nThis convention allows us to write:\nindent=1\nWith Gödel's convention, the axiom of limitation of size can be written:\n\n# Implications of the axiom\nVon Neumann proved that the axiom of limitation of size implies the [axiom of replacement](https://bliptext.com/articles/axiom-of-replacement), which can be expressed as: If F is a function and A is a set, then F(A) is a set.\nThis is [proved by contradiction](https://bliptext.com/articles/proof-by-contradiction).\nLet F be a function and A be a set.\nAssume that F(A) is a proper class.\nThen there is a function G that maps F(A) onto V.\nSince the [composite function](https://bliptext.com/articles/composite-function) 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.\nTherefore, F(A) is a set.\nSince the [axiom of replacement implies the axiom of separation](https://bliptext.com/articles/axiom-of-replacement), the axiom of limitation of size implies the [axiom of separation](https://bliptext.com/articles/axiom-of-separation).\nVon Neumann also proved that his axiom implies that V can be [well-ordered](https://bliptext.com/articles/well-ordered).\nThe proof starts by proving by contradiction that Ord, the class of all [ordinals](https://bliptext.com/articles/ordinal-mathematics), is a proper class.\nAssume that Ord is a set.\nSince it is a [transitive set](https://bliptext.com/articles/transitive-set) that is strictly well-ordered by ∈, it is an ordinal.\nSo Ord ∈ Ord, which contradicts Ord being strictly well-ordered by ∈.\nTherefore, Ord is a proper class.\nSo von Neumann's axiom implies that there is a function F that maps Ord onto V.\nTo 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 = {(α, x) ∈ F: ∀β((β, x) ∈ F ⇒ α ≤ β)}.\nThe function G is a [one-to-one correspondence](https://bliptext.com/articles/one-to-one-correspondence) between a subset of Ord and V.\nTherefore, x < y if G−1(x) < G−1(y) defines a well-ordering of V.\nThis well-ordering defines a global [choice function](https://bliptext.com/articles/choice-function): Let Inf(x) be the least element of a non-empty set x.\nSince Inf(x) ∈ x, this function chooses an element of x for every non-empty set x.\nTherefore, Inf(x) is a global choice function, so Von Neumann's axiom implies the [axiom of global choice](https://bliptext.com/articles/axiom-of-global-choice).\nIn 1968, [Azriel Lévy](https://bliptext.com/articles/azriel-l-vy) proved that von Neumann's axiom implies the [axiom of union](https://bliptext.com/articles/axiom-of-union).\nFirst, he proved without using the axiom of union that every set of ordinals has an upper bound.\nThen he used a function that maps Ord onto V to prove that if A is a set, then ∪A is a set.\nThe axioms of replacement, global choice, and union (with the other axioms of [NBG](https://bliptext.com/articles/von-neumann-bernays-g-del-set-theory)) imply the axiom of limitation of size.\nTherefore, this axiom is equivalent to the combination of replacement, global choice, and union in NBG or [Morse–Kelley set theory](https://bliptext.com/articles/morse-kelley-set-theory).\nThese 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.\nLévy's proof that this axiom is redundant came many years later.\nThe axioms of NBG with the axiom of global choice replaced by the usual [axiom of choice](https://bliptext.com/articles/axiom-of-choice) do not imply the axiom of limitation of size.\nIn 1964, [William B. Easton](https://bliptext.com/articles/william-bigelow-easton) used [forcing](https://bliptext.com/articles/forcing-set-theory) to build a [model](https://bliptext.com/articles/model-theory) of NBG with global choice replaced by the axiom of choice.\nIn Easton's model, V cannot be [linearly ordered](https://bliptext.com/articles/linearly-ordered), so it cannot be well-ordered.\nTherefore, the axiom of limitation of size fails in this model.\nOrd 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.\nThe axioms of NBG with the axiom of replacement replaced by the weaker axiom of separation do not imply the axiom of limitation of size.\nDefine as the \\alpha-th infinite [initial ordinal](https://bliptext.com/articles/initial-ordinal), which is also the [cardinal](https://bliptext.com/articles/cardinal-number) ; numbering starts at 0, so In 1939, Gödel pointed out that Lω ω, a subset of the [constructible universe](https://bliptext.com/articles/constructible-universe), is a model of [ZFC](https://bliptext.com/articles/zfc) with replacement replaced by separation.\nTo expand it into a model of NBG with replacement replaced by separation, let its classes be the sets of Lω ω+1, which are the constructible subsets of Lω ω.\nThis model satisfies NBG's class existence axioms because restricting the set variables of these axioms to Lω ω produces [instances](https://bliptext.com/articles/instance-predicate-logic) of the axiom of separation, which holds in L. It satisfies the axiom of global choice because there is a function belonging to Lω ω+1 that maps ωω onto Lω ω, which implies that Lω ω is well-ordered.\nThe axiom of limitation of size fails because the proper class {ωn : n ∈ ω} has cardinality [\\aleph 0](https://bliptext.com/articles/aleph-number), so it cannot be mapped onto Lω ω, which has cardinality.\nIn 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.\nThe axiom of limitation of size implies von Neumann's 1923 axiom.\nTherefore, it also implies that all proper classes are [equinumerous](https://bliptext.com/articles/equinumerous) with V.\n\n# Zermelo's models and the axiom of limitation of size\nIn 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.\nThese models are built in [ZFC](https://bliptext.com/articles/zfc) by using the [cumulative hierarchy](https://bliptext.com/articles/cumulative-hierarchy) Vα, which is defined by [transfinite recursion](https://bliptext.com/articles/transfinite-recursion):\nZermelo worked with models of the form Vκ where κ is a [cardinal](https://bliptext.com/articles/von-neumann-cardinal).\nThe classes of the model are the [subsets](https://bliptext.com/articles/subsets) of Vκ, and the model's ∈-relation is the standard ∈-relation.\nThe sets of the model are the classes X such that X ∈ Vκ.\nZermelo identified cardinals κ such that Vκ satisfies:\nSince every class is a subset of Vκ, Theorem 2 implies that every class X has [cardinality](https://bliptext.com/articles/cardinality) ≤ κ.\nCombining this with Theorem 1 proves: every proper class has cardinality κ.\nHence, every proper class can be put into one-to-one correspondence with Vκ.\nThis correspondence is a subset of Vκ, so it is a class of the model.\nTherefore, the axiom of limitation of size holds for the model Vκ.\nThe theorem stating that Vκ has a well-ordering can be [proved directly](https://bliptext.com/articles/direct-proof).\nSince κ is an ordinal of cardinality κ and |Vκ| = κ, there is a [one-to-one correspondence](https://bliptext.com/articles/one-to-one-correspondence) between κ and Vκ.\nThis correspondence produces a well-ordering of Vκ.\nVon Neumann's proof is [indirect](https://bliptext.com/articles/indirect-proof).\nIt uses the [Burali-Forti paradox](https://bliptext.com/articles/burali-forti-paradox) to prove by contradiction that the class of all ordinals is a proper class.\nHence, 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.\nThis function produces a well-ordering of Vκ.\n# [Ernst Zermelo in the 1900s | upload.wikimedia.org/wikipedia/commons/e/ed/Ernst///Zermelo///1900s.jpg]\n\n## The model Vω\nTo demonstrate that Theorems 1 and 2 hold for some Vκ, we first prove that if a set belongs to Vα then it belongs to all subsequent Vβ, or equivalently: Vα ⊆ Vβ for α ≤ β.\nThis is proved by [transfinite induction](https://bliptext.com/articles/transfinite-induction) on β:\nSets enter the cumulative hierarchy through the power set P(Vβ) at step β+1.\nThe following definitions will be needed:\nZermelo's smallest model is Vω.\n[Mathematical induction](https://bliptext.com/articles/mathematical-induction) proves that Vn is [finite](https://bliptext.com/articles/finite-set) for all n < ω:\nProof of Theorem 1: A set X enters Vω through P(Vn) for some n < ω, so X ⊆ Vn. Since Vn is finite, X is finite.\n[Conversely](https://bliptext.com/articles/converse-logic): If a class X is finite, let N = sup {rank(x): x ∈ X}.\nSince rank(x) ≤ N for all x ∈ X, we have X ⊆ VN+1, so X ∈ VN+2 ⊆ Vω.\nTherefore, X ∈ Vω.\nProof of Theorem 2: Vω is the union of [countably infinitely](https://bliptext.com/articles/countably-infinite) many finite sets of increasing size.\nHence, it has cardinality \\aleph_0, which equals ω by [von Neumann cardinal assignment](https://bliptext.com/articles/von-neumann-cardinal-assignment).\nThe sets and classes of Vω satisfy all the axioms of NBG except the [axiom of infinity](https://bliptext.com/articles/axiom-of-infinity).\n\n## The models Vκ where κ is a strongly inaccessible cardinal\nTwo properties of finiteness were used to prove Theorems 1 and 2 for Vω:\nTo find models satisfying the axiom of infinity, replace \"finite\" by \"< κ\" to produce the properties that define [strongly inaccessible cardinals](https://bliptext.com/articles/strongly-inaccessible-cardinal).\nA cardinal κ is strongly inaccessible if κ > ω and:\nThese properties assert that κ cannot be reached from below.\nThe first property says κ cannot be reached by power sets; the second says κ cannot be reached by the axiom of replacement.\nJust as the axiom of infinity is required to obtain ω, an axiom is needed to obtain strongly inaccessible cardinals.\nZermelo postulated the existence of an unbounded sequence of strongly inaccessible cardinals.\nIf κ is a strongly inaccessible cardinal, then transfinite induction proves |Vα| < κ for all α < κ:\nProof of Theorem 1: A set X enters Vκ through P(Vα) for some α < κ, so X ⊆ Vα.\nSince |Vα| < κ, we obtain |X| < κ.\nConversely: If a class X has |X| < κ, let β = sup {rank(x): x ∈ X}.\nBecause κ is strongly inaccessible, |X| < κ and rank(x) < κ for all x ∈ X imply β = sup {rank(x): x ∈ X} < κ.\nSince rank(x) ≤ β for all x ∈ X, we have X ⊆ Vβ+1, so X ∈ Vβ+2 ⊆ Vκ.\nTherefore, X ∈ Vκ.\nProof of Theorem 2: |Vκ| = |∪α < κ Vα| ≤ sup {|Vα| : α < κ}.\nLet β be this supremum.\nSince each ordinal in the supremum is less than κ, we have β ≤ κ.\nAssume β < κ.\nThen there is a cardinal λ such that β < λ < κ; for example, let λ = 2undefined.\nSince λ ⊆ Vλ and |Vλ| is in the supremum, we have λ ≤ |Vλ| ≤ β.\nThis contradicts β < λ.\nTherefore, |Vκ| = β = κ.\nThe sets and classes of Vκ satisfy all the axioms of NBG.\n\n# Limitation of size doctrine\nThe limitation of size doctrine is a [heuristic](https://bliptext.com/articles/heuristic) principle that is used to justify axioms of set theory.\nIt avoids the set theoretical paradoxes by restricting the full (contradictory) comprehension axiom schema:\nto instances \"that do not give sets 'too much bigger' than the ones they use.\"\nIf \"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.\nThe axiom of replacement produces an image set f(x) that is not bigger than x.\nThe 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.\nThe axiom of choice produces a choice set whose size is not bigger than the size of the given set of nonempty sets.\nThe limitation of size doctrine does not justify the axiom of infinity:\nwhich uses the [empty set](https://bliptext.com/articles/empty-set) and sets obtained from the [empty set](https://bliptext.com/articles/empty-set) by iterating the [ordinal successor operation](https://bliptext.com/articles/successor-ordinal).\nSince these sets are finite, any set satisfying this axiom, such as ω, is much bigger than these sets.\nFraenkel and Lévy regard the empty set and the infinite set of [natural numbers](https://bliptext.com/articles/natural-numbers), whose existence is implied by the axioms of infinity and separation, as the starting point for generating sets.\nVon Neumann's approach to limitation of size uses the axiom of limitation of size.\nAs mentioned in, von Neumann's axiom implies the axioms of separation, replacement, union, and choice.\nLike Fraenkel and Lévy, von Neumann had to add the axiom of infinity to his system since it cannot be proved from his other axioms.\nThe differences between von Neumann's approach to limitation of size and Fraenkel and Lévy's approach are:\nThere is disagreement on whether the limitation of size doctrine justifies the power set axiom.\nMichael Hallett has analyzed the arguments given by Fraenkel and Lévy.\nSome of their arguments measure size by criteria other than cardinal size—for example, Fraenkel introduces \"comprehensiveness\" and \"extendability.\"\nHallett points out what he considers to be flaws in their arguments.\nHallett 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.\nThis 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.\nFor the case where size is measured by cardinal size, Hallett mentions [Paul Cohen](https://bliptext.com/articles/paul-cohen)'s work.\nStarting with a model of ZFC and, Cohen built a model in which the cardinality of the power set of ω is if the [cofinality](https://bliptext.com/articles/cofinality) of is not ω; otherwise, its cardinality is.\nSince 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).\nHallett 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.\"\nHe 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.\nHe also quotes [John L. Bell](https://bliptext.com/articles/john-lane-bell) and [Moshé Machover](https://bliptext.com/articles/mosh-machover): \"... the power set P(u) of a given [infinite] set u is proportional not only to the size of u but also to the 'richness' of the entire universe ...\"\nAfter 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).\"\nHallett considers the limitation of size doctrine valuable for justifying most of the axioms of set theory.\nHis arguments only indicate that it cannot justify the axioms of infinity and power set.\nHe concludes that \"von Neumann's explicit assumption [of the smallness of power-sets] seems preferable to Zermelo's, Fraenkel's, and Lévy's obscurely hidden implicit assumption of the smallness of power-sets.\"\n\n# History\nVon Neumann developed the axiom of limitation of size as a new method of identifying sets.\n[ZFC](https://bliptext.com/articles/zermelo-fraenkel-set-theory) identifies sets via its set building axioms.\nHowever, as [Abraham Fraenkel](https://bliptext.com/articles/abraham-fraenkel) pointed out: \"The rather arbitrary character of the processes which are chosen in the axioms of **Z** [**Z**FC] as the basis of the theory, is justified by the historical development of set-theory rather than by logical arguments.\"\nThe 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](https://bliptext.com/articles/well-ordering-theorem).\nIn 1922, Abraham Fraenkel and [Thoralf Skolem](https://bliptext.com/articles/thoralf-skolem) pointed out that [Zermelo's axioms](https://bliptext.com/articles/zermelo-set-theory) cannot prove the existence of the set {Z0, Z1, Z2, ...} where Z0 is the set of [natural numbers](https://bliptext.com/articles/natural-number), and Zn+1 is the power set of Zn. They also introduced the axiom of replacement, which guarantees the existence of this set.\nHowever, 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.\nIn 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.\nVon Neumann identified these sets using the criterion: \"A set is 'too big' if and only if it is [equivalent](https://bliptext.com/articles/equinumerous) with the set of all things.\"\nHe then restricted how these sets may be used: \"... in order to avoid the paradoxes those [sets] which are 'too big' are declared to be impermissible as elements.\"\nBy 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.\nBy 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.\nThis modification allowed von Neumann to give a simple proof of the axiom of replacement.\nVon Neumann's axiom identifies sets as classes that cannot be mapped onto V.\nVon Neumann realized that, even with this axiom, his set theory does not fully characterize sets.\nGödel found von Neumann's axiom to be \"of great interest\":","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-26T07:47:50.883Z","updated_at":"2025-01-26T07:47:50.883Z","search_vector":"'axiom':1 'limit':3 'size':5","search_content":" \n \n\n# \nIn [set theory](https://bliptext.com/articles/set-theory), the **axiom of limitation of size** was proposed by [John von Neumann](https://bliptext.com/articles/john-von-neumann) in his 1925 [axiom system](https://bliptext.com/articles/axiom-system) for [sets](https://bliptext.com/articles/set-mathematics) and [classes](https://bliptext.com/articles/class-set-theory).\nIt formalizes the [limitation of size](https://bliptext.com/articles/limitation-of-size) principle, which avoids the paradoxes encountered in earlier formulations of [set theory](https://bliptext.com/articles/set-theory) by recognizing that some classes are too big to be sets.\nVon Neumann realized that the paradoxes are caused by permitting these big classes to be members of a class.\nA class that is a member of a class is a set; a class that is not a set is a [proper class](https://bliptext.com/articles/proper-class).\nEvery class is a [subclass](https://bliptext.com/articles/subclass-set-theory) of [V](https://bliptext.com/articles/von-neumann-universe), the class of all sets.\nThe 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](https://bliptext.com/articles/onto) V.\nUsually, this axiom is stated in the [equivalent](https://bliptext.com/articles/logically-equivalent) form: A class is a proper class if and only if there is a function that maps it onto V.\nVon Neumann's axiom implies the axioms of [replacement](https://bliptext.com/articles/axiom-of-replacement), [separation](https://bliptext.com/articles/axiom-of-separation), [union](https://bliptext.com/articles/axiom-of-union), and [global choice](https://bliptext.com/articles/axiom-of-global-choice).\nIt is equivalent to the combination of replacement, union, and global choice in [Von Neumann–Bernays–Gödel set theory](https://bliptext.com/articles/von-neumann-bernays-g-del-set-theory) (NBG) and [Morse–Kelley set theory](https://bliptext.com/articles/morse-kelley-set-theory).\nLater expositions of class theories—such as those of [Paul Bernays](https://bliptext.com/articles/paul-bernays), [Kurt Gödel](https://bliptext.com/articles/kurt-g-del), and [John L. Kelley](https://bliptext.com/articles/john-l-kelley)—use replacement, union, and a choice axiom equivalent to global choice rather than von Neumann's axiom.\nIn 1930, [Ernst Zermelo](https://bliptext.com/articles/ernst-zermelo) defined models of set theory satisfying the axiom of limitation of size.\n[Abraham Fraenkel](https://bliptext.com/articles/abraham-fraenkel) and [Azriel Lévy](https://bliptext.com/articles/azriel-l-vy) 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](https://bliptext.com/articles/axiom-of-power-set).\n[Michael Hallett](https://bliptext.com/articles/michael-hallett) has argued that the limitation of size doctrine does not justify the power set axiom and that \"von Neumann's explicit assumption [of the smallness of power-sets] seems preferable to Zermelo's, Fraenkel's, and Lévy's obscurely hidden implicit assumption of the smallness of power-sets.\"\n\n# Formal statement\nThe 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](https://bliptext.com/articles/formal-language) of set theory as:\nGödel introduced the convention that uppercase variables range over all the classes, while lowercase variables range over all the sets.\nThis convention allows us to write:\nindent=1\nWith Gödel's convention, the axiom of limitation of size can be written:\n\n# Implications of the axiom\nVon Neumann proved that the axiom of limitation of size implies the [axiom of replacement](https://bliptext.com/articles/axiom-of-replacement), which can be expressed as: If F is a function and A is a set, then F(A) is a set.\nThis is [proved by contradiction](https://bliptext.com/articles/proof-by-contradiction).\nLet F be a function and A be a set.\nAssume that F(A) is a proper class.\nThen there is a function G that maps F(A) onto V.\nSince the [composite function](https://bliptext.com/articles/composite-function) 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.\nTherefore, F(A) is a set.\nSince the [axiom of replacement implies the [axiom of separation](https://bliptext.com/articles/axiom-of-separation)](https://bliptext.com/articles/axiom-of-replacement), the axiom of limitation of size implies the [axiom of separation](https://bliptext.com/articles/axiom-of-separation).\nVon Neumann also proved that his axiom implies that V can be [well-ordered](https://bliptext.com/articles/well-ordered).\nThe proof starts by proving by contradiction that Ord, the class of all [ordinals](https://bliptext.com/articles/ordinal-mathematics), is a proper class.\nAssume that Ord is a set.\nSince it is a [transitive set](https://bliptext.com/articles/transitive-set) that is strictly well-ordered by ∈, it is an ordinal.\nSo Ord ∈ Ord, which contradicts Ord being strictly well-ordered by ∈.\nTherefore, Ord is a proper class.\nSo von Neumann's axiom implies that there is a function F that maps Ord onto V.\nTo 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 = {(α, x) ∈ F: ∀β((β, x) ∈ F ⇒ α ≤ β)}.\nThe function G is a [one-to-one correspondence](https://bliptext.com/articles/one-to-one-correspondence) between a subset of Ord and V.\nTherefore, x ω and:\nThese properties assert that κ cannot be reached from below.\nThe first property says κ cannot be reached by power sets; the second says κ cannot be reached by the axiom of replacement.\nJust as the axiom of infinity is required to obtain ω, an axiom is needed to obtain strongly inaccessible cardinals.\nZermelo postulated the existence of an unbounded sequence of strongly inaccessible cardinals.\nIf κ is a strongly inaccessible cardinal, then transfinite induction proves |Vα| < κ for all α < κ:\nProof of Theorem 1: A set X enters Vκ through P(Vα) for some α < κ, so X ⊆ Vα.\nSince |Vα| < κ, we obtain |X| < κ.\nConversely: If a class X has |X| < κ, let β = sup {rank(x): x ∈ X}.\nBecause κ is strongly inaccessible, |X| < κ and rank(x) < κ for all x ∈ X imply β = sup {rank(x): x ∈ X} < κ.\nSince rank(x) ≤ β for all x ∈ X, we have X ⊆ Vβ+1, so X ∈ Vβ+2 ⊆ Vκ.\nTherefore, X ∈ Vκ.\nProof of Theorem 2: |Vκ| = |∪α < κ Vα| ≤ sup {|Vα| : α < κ}.\nLet β be this supremum.\nSince each ordinal in the supremum is less than κ, we have β ≤ κ.\nAssume β < κ.\nThen there is a cardinal λ such that β < λ < κ; for example, let λ = 2undefined.\nSince λ ⊆ Vλ and |Vλ| is in the supremum, we have λ ≤ |Vλ| ≤ β.\nThis contradicts β < λ.\nTherefore, |Vκ| = β = κ.\nThe sets and classes of Vκ satisfy all the axioms of NBG.\n\n# Limitation of size doctrine\nThe limitation of size doctrine is a [heuristic](https://bliptext.com/articles/heuristic) principle that is used to justify axioms of set theory.\nIt avoids the set theoretical paradoxes by restricting the full (contradictory) comprehension axiom schema:\nto instances \"that do not give sets 'too much bigger' than the ones they use.\"\nIf \"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.\nThe axiom of replacement produces an image set f(x) that is not bigger than x.\nThe 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.\nThe axiom of choice produces a choice set whose size is not bigger than the size of the given set of nonempty sets.\nThe limitation of size doctrine does not justify the axiom of infinity:\nwhich uses the [empty set](https://bliptext.com/articles/empty-set) and sets obtained from the [empty set](https://bliptext.com/articles/empty-set) by iterating the [ordinal successor operation](https://bliptext.com/articles/successor-ordinal).\nSince these sets are finite, any set satisfying this axiom, such as ω, is much bigger than these sets.\nFraenkel and Lévy regard the empty set and the infinite set of [natural numbers](https://bliptext.com/articles/natural-numbers), whose existence is implied by the axioms of infinity and separation, as the starting point for generating sets.\nVon Neumann's approach to limitation of size uses the axiom of limitation of size.\nAs mentioned in, von Neumann's axiom implies the axioms of separation, replacement, union, and choice.\nLike Fraenkel and Lévy, von Neumann had to add the axiom of infinity to his system since it cannot be proved from his other axioms.\nThe differences between von Neumann's approach to limitation of size and Fraenkel and Lévy's approach are:\nThere is disagreement on whether the limitation of size doctrine justifies the power set axiom.\nMichael Hallett has analyzed the arguments given by Fraenkel and Lévy.\nSome of their arguments measure size by criteria other than cardinal size—for example, Fraenkel introduces \"comprehensiveness\" and \"extendability.\"\nHallett points out what he considers to be flaws in their arguments.\nHallett 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.\nThis 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.\nFor the case where size is measured by cardinal size, Hallett mentions [Paul Cohen](https://bliptext.com/articles/paul-cohen)'s work.\nStarting with a model of ZFC and, Cohen built a model in which the cardinality of the power set of ω is if the [cofinality](https://bliptext.com/articles/cofinality) of is not ω; otherwise, its cardinality is.\nSince 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).\nHallett 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.\"\nHe 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.\nHe also quotes [John L. Bell](https://bliptext.com/articles/john-lane-bell) and [Moshé Machover](https://bliptext.com/articles/mosh-machover): \"... the power set P(u) of a given [infinite] set u is proportional not only to the size of u but also to the 'richness' of the entire universe ...\"\nAfter 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).\"\nHallett considers the limitation of size doctrine valuable for justifying most of the axioms of set theory.\nHis arguments only indicate that it cannot justify the axioms of infinity and power set.\nHe concludes that \"von Neumann's explicit assumption [of the smallness of power-sets] seems preferable to Zermelo's, Fraenkel's, and Lévy's obscurely hidden implicit assumption of the smallness of power-sets.\"\n\n# History\nVon Neumann developed the axiom of limitation of size as a new method of identifying sets.\n[ZFC](https://bliptext.com/articles/zermelo-fraenkel-set-theory) identifies sets via its set building axioms.\nHowever, as [Abraham Fraenkel](https://bliptext.com/articles/abraham-fraenkel) pointed out: \"The rather arbitrary character of the processes which are chosen in the axioms of **Z** [**Z**FC] as the basis of the theory, is justified by the historical development of set-theory rather than by logical arguments.\"\nThe 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](https://bliptext.com/articles/well-ordering-theorem).\nIn 1922, Abraham Fraenkel and [Thoralf Skolem](https://bliptext.com/articles/thoralf-skolem) pointed out that [Zermelo's axioms](https://bliptext.com/articles/zermelo-set-theory) cannot prove the existence of the set {Z0, Z1, Z2, ...} where Z0 is the set of [natural numbers](https://bliptext.com/articles/natural-number), and Zn+1 is the power set of Zn. They also introduced the axiom of replacement, which guarantees the existence of this set.\nHowever, 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.\nIn 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.\nVon Neumann identified these sets using the criterion: \"A set is 'too big' if and only if it is [equivalent](https://bliptext.com/articles/equinumerous) with the set of all things.\"\nHe then restricted how these sets may be used: \"... in order to avoid the paradoxes those [sets] which are 'too big' are declared to be impermissible as elements.\"\nBy 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.\nBy 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.\nThis modification allowed von Neumann to give a simple proof of the axiom of replacement.\nVon Neumann's axiom identifies sets as classes that cannot be mapped onto V.\nVon Neumann realized that, even with this axiom, his set theory does not fully characterize sets.\nGödel found von Neumann's axiom to be \"of great interest\":","revisionCount":0},"hasMore":false,"nextCursor":null}