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2014-01-30T19:11:47Z

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In class theories, the '''axiom of limitation of size''' says that for any class ''C'', ''C'' is a [[proper class]], that is a class which is not a [[Set (mathematics)|set]] (an [[Element (mathematics)|element]] of other classes), if and only if it can be mapped [[onto]] the class ''[[Von Neumann universe|V]]'' of all sets.<ref>This is roughly von Neumann's original formulation, see Fraenkel & al, p. 137.</ref>

:<math>\forall C [\lnot \exist W (C \in W) \iff \exist F ( \forall x [\exist W (x \in W) \Rightarrow \exist s (s \in C \and \langle s, x \rangle \in F)] \and </math>
::<math>\forall x \forall y \forall s [(\langle s, x \rangle \in F \and \langle s, y \rangle \in F) \Rightarrow x = y])].</math>

This axiom is due to [[John von Neumann]]. It implies the [[axiom schema of specification]], [[axiom schema of replacement]], [[axiom of global choice]], and even, as noticed later by [[Azriel Levy]], [[axiom of union]]<ref>showing directly that a set of ordinals has an upper bound, see A. Levy, " On von Neumann's axiom system for set theory ", Amer. Math. Monthly, 75 (1968), p. 762-763.</ref> at one stroke. The axiom of limitation of size implies the axiom of global choice because the class of ordinals is not a set, so there is a [[surjection]] from the [[ordinal number|ordinals]] to the [[Von Neumann universe|universe]], thus an [[injection (mathematics)|injection]] from the universe to the ordinals, that is, the universe of sets is [[well-order]]ed.

Together the [[axiom of replacement]] and the [[axiom of global choice]] (with the other axioms of [[von Neumann–Bernays–Gödel set theory]]) imply this axiom. This axiom can then replace replacement, global choice, specification and union in von Neumann–Bernays–Gödel or [[Morse–Kelley set theory]].

However, the axiom of replacement and the usual [[axiom of choice]] (with the other axioms of von Neumann–Bernays–Gödel set theory) do not imply von Neumann's axiom. In 1964, Easton used [[Forcing (set theory)|forcing]] to build a [[Model theory|model]] that satisfies the axioms of von Neumann–Bernays–Gödel set theory with one exception: the axiom of global choice is replaced by the axiom of choice. In Easton's model, the axiom of limitation of size fails dramatically: the universe of sets cannot even be [[linearly ordered]].<ref>Easton 1964.</ref>

It can be shown that a class is a proper class if and only if it is [[equinumerous]] to ''V'', but von Neumann's axiom does not capture all of the "[[limitation of size]] doctrine",<ref>Fraenkel & al, p. 137. A guiding principle for ZF to avoid set theoretical paradoxes is to restrict to instances of full (contradictory) comprehension scheme that do not give sets "too much bigger" than the ones they use; it is known as "limitation of size", Fraenkel & al call it "limitation of size doctrine", see p. 32.</ref> because the [[axiom of power set]] is not a consequence of it. Later expositions of class theories ([[Paul Bernays|Bernays]], [[Kurt Gödel|Gödel]], [[John L. Kelley|Kelley]], ...) generally use replacement and a form of the axiom of choice rather than the axiom of limitation of size.

==History==

Von Neumann developed the axiom of limitation of size as a new method of identifying sets. [[Zermelo–Fraenkel set theory|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''' [ZFC] as the basis of the theory, is justified by the historical development of set-theory rather than by logical arguments."<ref>''Historical Introduction'' in Bernays 1991, p. 31.</ref>

The historical development of the ZFC axioms began in 1908 when [[Ernst Zermelo|Zermelo]] chose axioms to support his proof of the [[well-ordering theorem]] and to avoid contradictory sets.<ref>"... we must, on the one hand, restrict these principles [axioms] sufficiently to exclude all contradictions and, on the other hand, take them sufficiently wide to retain all that is valuable in this theory." (Zermelo 1908, p. 261; English translation, p. 200). Gregory Moore analyzed Zermelo's reasons behind his axiomatization and concluded that "his axiomatization was primarily motivated by a desire to secure his demonstration of the Well-Ordering Theorem …" and "For Zermelo, … the paradoxes were an inessential obstacle to be circumvented with as little fuss as possible." (Moore 1982, p. 159–160).</ref> In 1922, [[Abraham Fraenkel|Fraenkel]] and [[Thoralf Skolem|Skolem]] pointed out that [[Zermelo set theory|Zermelo's axioms]] cannot prove the existence of the set {''Z''0, ''Z''1, ''Z''2, … } where ''Z''0 is the set of [[natural number]]s, and ''Z''''n''+1 is the [[power set]] of ''Z''''n''.<ref>Frankel 1922, p. 230–231; Skolem 1922 (English translation, p. 296–297).</ref> They also introduced the [[axiom of replacement]], which guarantees the existence of this set.<ref>Ferreirós 2007, p. 369. In 1917, [[Dmitry Mirimanoff|Mirimanoff]] published a form of replacement based on cardinal equivalence (Mirimanoff 1917, p. 49).</ref> 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 the sets that are "too big" (now called proper classes) and that can lead to contradictions.<ref>He gave a detailed exposition of his set theory in two articles: von Neumann 1925 and von Neumann 1928.</ref> Von Neumann identified these sets using the criterion: "A set is 'too big' if and only if it is [[equinumerous|equivalent]] to the set of all things."<ref>Hallett 1984, p. 288.</ref> He 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''."<ref>Hallett 1984, p. 290.</ref> By combining this restriction with his criterion, von Neumann obtained the axiom of limitation of size (which in the language of classes states): A class X is not an element of any class if and only if X is equivalent to the class of all sets.<ref>Hallett 1984, p. 290. Von Neumann later changed "equivalent to the class of all sets" to "can be mapped onto the class of all sets."</ref> So von Neumann identified sets as classes that are not equivalent to the class of all sets. Von Neumann realized that, even with his new axiom, his set theory does not fully characterize sets.<ref>To be precise, von Neumann investigated whether his set theory is [[categorical (model theory)|categorical]]; that is, whether it uniquely determines sets in the sense that any two of its models are [[isomorphic]]. He showed that it is not categorical because of a weakness in the [[axiom of regularity]]: this axiom only excludes descending ∈-sequences from existing in the model; descending sequences may still exist outside the model. A model having "external" descending sequences is not isomorphic to a model having no such sequences since this latter model lacks isomorphic images for the sets belonging to external descending sequences. This led von Neumann to conclude "that no categorical axiomatization of set theory seems to exist at all" (von Neumann 1925, p. 239; English translation: p. 412).</ref>

Gödel found von Neumann's axiom to be "of great interest":

:"In particular I believe that his [von Neumann's] necessary 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.<ref>For example, von Neumann's proof that his axiom implies the well-ordering theorem uses the [[Burali-Forte paradox]] (von Neumann 1925, p. 223; English translation: p. 398).</ref> Moreover I believe that only by going farther in this direction, i.e., in the direction opposite to [[constructivism (mathematics)|constructivism]], will the basic problems of abstract set theory be solved."<ref>From a Nov. 8, 1957 letter Gödel wrote to [[Stanislaw Ulam]] (Kanamori 2003, p. 295).</ref>

==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]] ''V''α, which is defined by [[transfinite recursion]]:
# ''V''0 = [[empty set|∅]].<ref>This is the standard definition of ''V''0. Zermelo let ''V''0 be a set of [[urelements]] and proved that if this set contains a single element, the resulting model satisfies the axiom of limitation of size (his proof also works for ''V''0 = ∅). Zermelo stated that the axiom is not true for all models built from a set of urelements. (Zermelo 1930, p. 38; English translation: p. 1227.)</ref>
# ''V''α+1 = ''V''α ∪ ''P''(''V''α). That is, the [[Union (set theory)|union]] of ''V''α and its [[power set]].<ref>This is Zermelo's definition (Zermelo 1930, p. 36; English translation: p. 1225 & p. 1209), which is equivalent to ''V''α+1 = ''P''(''V''α) since ''V''α &sube; ''P''(''V''α) (Kunen 1980, p. 95; Kunen uses the notation R(α) instead of ''V''α).</ref>
# 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 [[von Neumann cardinal|cardinal]]. The classes of the model are the [[subsets]] of ''V''κ, and the model's ∈-relation is the standard ∈-relation. The sets of the model ''V''κ are the classes ''X'' such that ''X'' ∈ ''V''κ.<ref>In [[von Neumann–Bernays–Gödel set theory|NBG]], ''X'' is a set if there is a class ''Y'' such that ''X'' ∈ ''Y''. Since ''Y'' &sube; ''V''κ, we have ''X'' ∈ ''V''κ. Conversely, if ''X'' ∈ ''V''κ, then ''X'' belongs to a class, so ''X'' is a set.</ref> Zermelo identified cardinals κ such that ''V''κ satisfies:<ref>These theorems are part of Zermelo's Second Development Theorem. (Zermelo 1930, p. 37; English translation: p. 1226.)</ref>
: 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]] ≤ 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, so the axiom of limitation of size holds for the model ''V''κ.

The proof of the axiom of global choice in ''V''κ is more direct than von Neumann's proof. First note that κ (being a [[von Neumann cardinal]]) is a [[well-ordered]] class of cardinality κ. Since Theorem 2 states that ''V''κ has cardinality κ, there is a [[one-to-one correspondence]] between κ and ''V''κ. This correspondence produces a well-ordering of ''V''κ, which implies the axiom of global choice.<ref>The domain of the global [[choice function]] consists of the non-empty sets of ''V''κ; this function uses the well-ordering of ''V''κ to choose the least element of each set.</ref> Von Neumann uses the [[Burali-Forti paradox]] to [[prove by contradiction]] that the class of all ordinals is a proper class, and then he applies the axiom of limitation of size to well-order the universal class.<ref>Von Neumann 1925, p. 223. English translation: p. 398. Von Neumann's proof, which only uses axioms, has the advantage of applying to all models rather than just to ''V''κ.</ref>

===The model ''V''ω===

To demonstrate that Theorems 1 and 2 hold for some ''V''κ, we need to prove that if a set belongs to ''V''α then it belongs to all subsequent ''V''β, or equivalently: ''V''α &sube; ''V''β for α ≤ β. This is proved by [[transfinite induction]] on β:
# β = 0: ''V''0 &sube; ''V''0.
# For β+1: By inductive hypothesis, ''V''α &sube; ''V''β. Hence, ''V''α &sube; ''V''β &sube; ''V''β ∪ ''P''(''V''β) = ''V''β+1.
# For limit β: If α < β, then ''V''α &sube; ∪ξ < β ''V''ξ = ''V''β. If α = β, then ''V''α &sube; ''V''β.
Note that sets enter the hierarchy only through the power set ''P''(''V''β) at step β+1. We will need the following definitions:
:If ''x'' is a set, '''rank'''(''x'') is the least ordinal β such that ''x'' ∈ ''V''β+1.<ref>Kunen 1980, p. 95.</ref>
:The '''supremum''' 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|Induction]] proves that ''V''''n'' is [[finite set|finite]] 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.

To prove Theorem 1: since a set ''X'' enters ''V''ω only through ''P''(''V''''n'') for some ''n'' < ω, we have ''X'' &sube; ''V''''n''. Since ''V''''n'' is finite, ''X'' is finite. [[Converse (logic)|Conversely]]: if a class ''X'' is finite, let ''N'' = sup {rank(''x''): ''x'' ∈ ''X''}. Since rank(''x'') ≤ ''N'' for all ''x'' ∈ ''X'', we have ''X'' &sube; ''V''''N''+1, so ''X'' ∈ ''V''''N''+2 &sube; ''V''ω. Therefore, ''X'' ∈ ''V''ω.

To prove Theorem 2, note that ''V''ω is the union of [[countably many]] finite sets. Hence, ''V''ω is countably infinite and has cardinality <math>\aleph_0</math> (which equals ω by [[von Neumann cardinal assignment]]).

It can be shown that the sets and classes of ''V''ω satisfy all the axioms of NBG (von Neumann–Bernays–Gödel set theory) except the [[axiom of infinity]].

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

To find models satisfying the axiom of infinity, observe that 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.
Replacing "finite" by "< κ" produces the properties that define [[strongly inaccessible cardinal]]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.<ref>Zermelo introduced strongly inaccessible cardinals κ so that ''V''κ would satisfy ZFC. The axioms of power set and replacement led him to the properties of strongly inaccessible cardinals. (Zermelo 1930, p. 31–35; English translation: p. 1221–1224.) Independently, [[Wacław Sierpiński|Sierpiński]] and [[Alfred Tarski|Tarski]] also introduced these cardinals in 1930.</ref> 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.<ref>Zermelo used this sequence of cardinals to obtain a sequence of models that explains the paradoxes of set theory — such as, the Burali-Forti paradox and [[Russell's paradox]]. He stated that the paradoxes "depend solely on confusing ''set theory itself'' … with individual ''models'' representing it. What appears as an 'ultrafinite non- or super-set' in one model is, in the succeeding model, a perfectly good, valid set with both a cardinal number and an ordinal type, and is itself a foundation stone for the construction of a new domain [model]." (Zermelo 1930, p. 46–47; English translation: p. 1233.)</ref>

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 {| ''V''ξ | : ξ < α} < κ. Last inequality uses inductive hypothesis and κ being strongly inaccessible.

To prove Theorem 1: since a set ''X'' enters ''V''κ only through ''P''(''V''α) for some α < κ, we have ''X'' &sube; ''V''α. Since | ''V''α | < κ, we have | ''X'' | < κ. Conversely: if a class ''X'' has | ''X'' | < κ, let β = sup {rank(''x''): ''x'' ∈ ''X''}. Since κ is strongly inaccessible, | ''X'' | < κ, and rank(''x'') < κ for all ''x'' ∈ ''X'', we have β < κ. Also, rank(''x'') ≤ β for all ''x'' ∈ ''X'' implies ''X'' &sube; ''V''β+1, so ''X'' ∈ ''V''β+2 &sube; ''V''κ. Therefore, ''X'' ∈ ''V''κ.

To prove Theorem 2, we compute: | ''V''κ | = | ∪α < κ ''V''α | ≤ sup {| ''V''α | : α < κ}. Let β be this supremum. Since each ordinal in the supremum is less than κ, we have β ≤ κ. Now β cannot be less than κ. If it were, there would be a cardinal λ such that β < λ < κ; for example, take λ = 2 | β |. Since λ &sube; ''V''λ and | ''V''λ | is in the supremum, we have λ ≤ | ''V''λ | ≤ β. This contradicts β < λ. Therefore, | ''V''κ | = β = κ.

It can be shown that the sets and classes of ''V''κ satisfy all the axioms of NBG.<ref>Zermelo proved that ZFC without the axiom of infinity is satisfied by ''V''κ for κ = ω and κ strongly inaccessible. To prove the class existence axioms of NBG (Gödel 1940, p. 5), note that ''V''κ is a set when viewed from the set theory that constructs it. Therefore, the [[axiom of specification]] produces subsets of ''V''κ that satisfy the class existence axioms.</ref>

==See also==
*[[Axiom of global choice]]
*[[Limitation of size]]
*[[Von Neumann–Bernays–Gödel set theory]]
*[[Morse–Kelley set theory]]

== Notes ==
<references/>

== References ==
* {{Citation | last = Bernays | first = Paul | authorlink = Paul Bernays | title=Axiomatic Set Theory | publisher=Dover Publications | year=1991 | isbn=0-486-66637-9}}.

* William B. Easton (1964), ''Powers of Regular Cardinals'', Ph.D. thesis, Princeton University.

* {{Citation | last = Ferreirós | first = José | title = Labyrinth of Thought: A History of Set Theory and Its Role in Mathematical Thought | place = Basel, Switzerland | publisher = Birkhäuser | year = 2007 | edition = 2nd revised | isbn = 3-7643-8349-6}}.

* {{Citation | last = Fraenkel | first = Abraham | title = Zu den Grundlagen der Cantor-Zermeloschen Mengenlehre | url = http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN235181684_0086&DMDID=DMDLOG_0019 | journal = [[Mathematische Annalen]] | volume = 86 | | pages = 230–237 | year = 1922}}.

* {{Citation | last1 = Fraenkel | first1 = Abraham | last2 = Bar-Hillel | first2 = Yehoshua | last3 = Levy | first3 = Azriel | title = Foundations of Set Theory | place = Basel, Switzerland | publisher = Elsevier | year = 1973 | edition = 2nd revised | isbn = 0-7204-2270-1}}.

* {{Citation | last = Gödel | first = Kurt | authorlink = Kurt Gödel | title = The Consistency of the Continuum Hypothesis | publisher = Princeton University Press | year = 1940}}.

* {{Citation | last = Kanamori | first = Akihiro | authorlink = Akihiro Kanamori | chapter = Stanislaw Ulam | url = http://math.bu.edu/people/aki/9.pdf}} in: {{Citation | author = Solomon Fefermann and John W. Dawson, Jr. (editors-in-chief) | title = Kurt Gödel Collected Works, Volume V, Correspondence H-Z | publisher = Clarendon Press | pages = 280–300 | year = 2003}}.

*{{Citation | last = Kunen | first = Kenneth | title = [[Set Theory: An Introduction to Independence Proofs]] | publisher = North-Holland | year = 1980 | isbn = 0-444-85401-0}}.* {{Citation | last = Hallett | first = Michael | title = Cantorian Set Theory and Limitation of Size | place = Oxford | publisher = Clarendon Press | year = 1984 | isbn = 0-444-86839-9}}.

* {{Citation | last1=Mirimanoff | first1=Dmitry | title=Les antinomies de Russell et de Burali-Forti et le probleme fondamental de la theorie des ensembles | url=http://retro.seals.ch/digbib/view?rid=ensmat-001:1917:19::9&id=hitlist | year = 1917 | journal=L'Enseignement Mathématique | volume=19 | pages=37–52}}.

* {{Citation | last = Moore| first = Gregory H. | title = Zermelo's Axiom of Choice: Its Origins, Development, and Influence | publisher = Springer | year = 1982 | isbn = 0-387-90670-3}}.

*{{Citation | last1=Sierpiński | first1=Wacław | last2=Tarski | first2=Alfred | title=Sur une propriété caractéristique des nombres inaccessibles | url=http://matwbn.icm.edu.pl/ksiazki/fm/fm15/fm15129.pdf | year=1930 | journal=[[Fundamenta Mathematicae]] | issn=0016-2736 | volume=15 | pages=292–300}}.

* {{Citation | last = Skolem | first = Thoralf | chapter = Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre | title = Matematikerkongressen i Helsingfors den 4-7 Juli, 1922 | pages = 217–232 | year = 1922}}. English translation: {{Citation | last=van Heijenoort | first =Jean | authorlink = Jean van Heijenoort | year =1967 | publisher = Harvard University Press | title = From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931 | chapter = Some remarks on axiomatized set theory | pages = 290–301 | isbn = 978-0-674-32449-7}}.

* {{Citation | last = von Neumann | first = John | title = Eine Axiomatisierung der Mengenlehre | url = http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN243919689_0154&DMDID=DMDLOG_0025 | journal = [[Journal für die Reine und Angewandte Mathematik]] | volume = 154 | pages = 219–240 | year = 1925}}. English translation: {{citation|last=van Heijenoort | first =Jean | year =1967 | publisher = Harvard University Press | title = From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931 | chapter = An axiomatization of set theory | pages = 393–413 | isbn = 978-0-674-32449-7}}.

* {{Citation | last = von Neumann | first = John | title = Die Axiomatisierung der Mengenlehre | url = http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN266833020_0027&DMDID=DMDLOG_0042 | journal = [[Mathematische Zeitschrift]] | volume = 27 | pages = 669–752 | year = 1928}}.

* {{Citation | last = Zermelo | first = Ernst | authorlink = Ernst Zermelo |title = Über Grenzzahlen und Mengenbereiche: neue Untersuchungen über die Grundlagen der Mengenlehre | url = http://matwbn.icm.edu.pl/ksiazki/fm/fm16/fm1615.pdf | journal = [[Fundamenta Mathematicae]] | volume = 16 | pages = 29–47| year = 1930}}. English translation: {{Citation | last = Ewald | first = William B. (ed.) | title = From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics | chapter = On boundary numbers and domains of sets: new investigations in the foundations of set theory | pages = 1208–1233 | publisher = Oxford University Press | year = 1996 | isbn = 978-0-19-853271-2}}.

* {{Citation|first=Ernst|last= Zermelo|year=1908|title=Untersuchungen über die Grundlagen der Mengenlehre I|journal=Mathematische Annalen |volume=65|issue=2|pages= 261–281|url = http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN235181684_0065&DMDID=DMDLOG_0018}}. English translation: {{citation|last=van Heijenoort | first =Jean | year =1967 | publisher = Harvard University Press | title = From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931 | chapter = Investigations in the foundations of set theory | pages = 199–215| isbn = 978-0-674-32449-7}}.
[[Category:Axioms of set theory]]
[[Category:Wellfoundedness]]

Total peripheral resistance

2013-07-11T21:09:07Z

68.109.180.117: to correct capitalization per MoS

{{Other uses|Deck (disambiguation){{!}}Deck}}
[[File:Wrau-olympic-deck.jpg|thumbnail|{{RMS|Olympic}}'s deck ]]
[[File:Falls of Clyde deck.jpg|right|thumb|The upper deck of the ''[[Falls of Clyde (ship)|Falls of Clyde]]'' is iron; a centre strip is planked with wood as a sort of walkway. As is typical for a late-19th-century vessel, several deckhouses may be seen.]]

A '''deck''' is a permanent covering over a [[Compartment (ship)|compartment]] or a [[hull (watercraft)|hull]]<ref>Edwards, Fred (illustrated by Sollers, Jim); ''Sailing as a Second Language: An illustrated dictionary''; International Marine Publishing Company; © 1988 Highmark Publishing Ltd.; ISBN 0-87742-965-0.</ref> of a [[ship]]. On a [[boat]] or [[ship]], the primary or upper deck is the horizontal structure which forms the 'roof' for the hull, which both strengthens the hull and serves as the primary working surface. Vessels often have more than one level both within the hull and in the superstructure above the primary deck which are similar to the floors of a multi-storey building, and which are also referred to as decks, as are specific compartments and decks built over specific areas of the superstructure. (Decks for some purposes have specific names; [[#Common names for decks|see below.]])

==Structure==
The purpose of the upper or primary deck is structural, and only secondarily to provide weather-tightness, and to support people and equipment. The deck serves as the lid to the complex box girder which is the hull. It resists tension, compression, and racking forces. The deck's [[scantling]] is usually the same as the [[topsides]], or might be heavier if the deck is expected to carry heavier loads (for example a [[container ship]]). The deck will be reinforced around deck fittings such as the [[Capstan (nautical)|capstan]], [[cleat (nautical)|cleat]]s, or [[bollard]]s.

[[File:QM2-wraparound-deck.jpg|thumb|Crew and passengers on the wraparound deck of [[RMS Queen Mary 2|RMS ''Queen Mary 2'']], an [[ocean liner]].]]

On ships with more than one level, deck refers to the level itself. The actual floor surface is called the sole, the term deck refers to a structural member tying the ships frames or ribs together over the keel. In modern ships, the interior decks are usually numbered from the primary deck, which is #1, downward and upward. So the first deck below the primary deck will be #2, and the first above the primary deck will be #A2 or #S2 (for "Above" or "Superstructure"). However, ships may also call decks by common names, or (especially on [[cruise ship]]s) may invent fanciful and romantic names for a specific deck or area of that specific ship, such as the ''[[Lido deck]]'' of the [[Princess Cruises]]' ''Love Boat''.

Equipment mounted on deck, such as the ship's wheel, [[binnacle]], [[fife rail]]s, and so forth, may be collectively referred to as deck furniture. Weather decks in western designs evolved from having structures fore and aft (forward or front and [[aft]] or The Rear of the ship mostly clear, then in the 19th century pilothouses/wheelhouses and deckhouses began to appear, eventually developing into the superstructure of modern ships. Eastern designs developed earlier, with efficient middle decks and minimalist fore and aft cabin structures across a range of designs.

== Common names for decks ==
[[File:Vasa-weather deck view.jpg|thumb|right|Weather deck of the Swedish 17th century warship ''[[Vasa (ship)|Vasa]]'' looking aft toward the sterncastle.]]
In vessels having more than one deck there are various naming conventions, numerically, alphabetically, etc. However, there are also various common historical names and types of decks:


{{anchor|Berth deck}}
* '''01 level''' is the term used in naval services to refer to the deck above the main deck. The next higher decks are referred to as the 02 level, the 03 level, and so on. Although these are formally called decks, they are usually referred to as levels, because they are usually incomplete decks that do not extend all the way from the [[stem (ship)|stem]] to the [[stern]] or across the ship. {{Citation needed|date=March 2012}}{{anchor|01 level}}
* '''Berth deck''': (''[[Naval]]'') A deck next below the gun deck, where the hammocks of the [[crew]] are slung.{{anchor|Boat deck}}
* '''Boat deck''': Especially on ships with [[sponson]]s, the deck area where lifeboats or the ship's [[Captain's Gig|gig]] are stored.{{anchor|Boiler deck}}
* '''Boiler deck''': (river steamers) The deck on which the [[boiler]]s are placed.{{anchor|Bridge deck}}
* '''Bridge deck''': (a) The deck area including the [[Ship's wheel|helm]] and navigation station, and where the [[Watchstanding|Officer of the Deck/Watch]] will be found, also known as the ''[[conn (nautical)|conn]]'' (b) An [[Glossary_of_nautical_terms#A|athwartships]] structure at the forward end of the [[Cockpit (sailing)|cockpit]] with a deck, often somewhat lower than the primary deck, to prevent a [[#Glossary|pooping wave]] from entering through the [[companionway]]. May also refer to the [[Deck (bridge)|deck]] of a [[bridge (nautical)|bridge]].
* '''[[Flight deck]]''': (''[[Naval]]'') A deck from which aircraft take off or land.
* '''[[Flush deck]]''': Any continuous unbroken deck from stem to stern.
* '''Forecastle deck''': The foremost part of the upper deck under which the sailors have their berths, extending from the foremast to the bow.
* '''[[Gun deck]]''': (''[[Naval]]'') a deck below the [[spar (sailing)|spar]] deck where the ships' cannon were carried. The term originally referred to a deck aboard a ship that was primarily used for the mounting of cannon to be fired in broadsides. However, on many smaller vessels such as frigates and unrated vessels the upper deck, forecastle and quarterdeck bore all of the cannons but were not referred to as the gun deck. The completely covered level under the upper deck was, however, still called the gun deck although it had no guns at all.
{{anchor|half-deck}}
* '''Half-deck''': That portion of the deck next below the spar deck which is between the [[mainmast]] and the [[cabin (ship)|cabin]].{{anchor|Helo deck}}
* '''[[Helicopter deck]]''' ('''heli deck'''): Usually located near the stern and always kept clear of obstacles hazardous to a [[helicopter]] landing.{{anchor|Hurricane deck}}
* '''Hurricane deck''': (River Steamers, etc.), the upper deck, usually a light deck, erected above the frame of the hull (deriving its name from the wind that always seemed to blow on the deck).<ref>[http://www.lakehistory.info/hrdeck.html Hurricane Deck]</ref>{{anchor|Main deck}}
* '''[[Lido deck]]''': Open area, typically at or near the stern of a passenger ship, housing the main outdoor swimming pool and sunbathing area.
* '''Lower deck''': (a) the deck immediately over the hold, orig. only of a ship with two decks.<ref>''[[Oxford English Dictionary]]''. "Lower n.4". Mar. 2009 Online edition. Retrieved 2009-04-06.</ref> (b) synonym for berth deck.
* '''[[Main deck]]''': The principal deck of a vessel; in some ships the highest deck of the hull, usually but not always the weather deck; in sailing warships often a deck under the upper deck.
* '''Middle or Waist deck''' The upper deck amidships, the working area of the deck.
* '''[[Orlop deck]]''': The deck or part of a deck where the [[cable]]s are stowed, usually below the [[waterline]]. It is the lowest deck in a ship.<ref>''[[Oxford English Dictionary]]''. "Orlop n.". Mar. 2009 Online edition. Retrieved 2009-04-06.</ref>
* '''[[Poop deck]]''': The deck forming the roof of a poop or poop cabin, built on the upper deck and extending from the [[mizzenmast]] aft.
* '''[[Promenade deck]]''': A "wrap-around porch" found on passenger ships and [[riverboat]]s encircling the superstructure. This can have open railings or be enclosed in glass, or a combination. Often the entire level where this is located is referred to as the Promenade Deck. {{anchor|Quarter-deck}}
* '''[[Quarterdeck]]''': (a) The part of the upper deck [[abaft]] the mainmast, including the poop deck when there is one. Usually reserved for ship's officers, guests, and passengers. (b) (''[[Naval]]'') The area to which a gangway for officers and diplomatic guests to board the vessel leads. Also any entry point for personnel.{{anchor|Side-deck}}
* '''[[Side-deck]]''': The upper deck outboard of any structures such as a coachroof or doghouse, also called a [[breezeway]]{{anchor|Spar deck}}
* '''Spar deck''': (a) Same as the upper deck. (b) Sometimes a light deck fitted over the upper deck.{{anchor|Sweep deck}}
* '''Sweep deck''': (''[[Naval]]'') The aftmost deck on a [[Minesweeper (ship)|minesweeper]], set close to the waterline for ease in launch and recovery of equipment.
* '''Topgallant forecastle deck''': Any raised deck occurring above the forecastle deck (see above).
* '''Tween deck''': the storage space between the hold and the main deck, often retractable.{{anchor|Upper deck}}
* '''Upper deck''': The highest deck of the hull, extending from [[Stem (ship)|stem]] to [[stern]].{{anchor|Weather deck}}
* '''Weather deck''': (a) Any deck exposed to the outside. (b) The [[windward]] side decks.<ref>Webster, Noah Ed.; ''Webster's Unabridged Dictionary'', 1913; [http://gutenberg.net/ Project Gutenberg] (eText numbers 660–670)</ref>
* '''[[Well deck]]''': (a) An exposed deck (weather deck) lower than decks fore and aft.<ref>{{cite web |url=http://www.uscg.mil/d8/sectumr/Prevention/docs/SPV_Guide.pdf |title=United States Coast Guard, Sector Upper Mississippi River, Small Passenger Vessel Information Package |author=United States Coast Guard |date= |work= |publisher=NARCIKI - Naval Architecture Wiki Project |accessdate=4 March 2012}}</ref> In particular it is one enclosed by bulwarks limiting flow of water and thus drainage.<ref>{{cite web |url=http://www.neely-chaulk.com/narciki/Well_deck%28merchant%29 |title=Well Deck (definition) |author= |date= |work= |publisher=NARCIKI - Naval Architecture Wiki Project |accessdate=4 March 2012}}</ref> (b) (''Modern naval'') A hangar-like deck located at the [[waterline]] in the stern of some amphibious assault ships, also known as a [[well dock]]. By taking on water the ship can lower the stern flooding the well deck and allowing boats and [[landing craft]] to dock within the ship.

== Construction ==

=== Methods in wood ===

A traditional [[wood]] deck would consist of planks laid [[#Glossary|fore and aft]] over [[#Glossary|beams]] and along [[#Glossary|carlins]], the seams of which are [[#Glossary|caulked]] and [[#Glossary|paid]] with [[#Glossary|tar]]. A yacht or other fancy boat might then have the deck canvased, with the [[Textile|fabric]] laid down in a thick layer of [[paint]] or sealant, and additional coats painted over. The wash or apron boards form the joint between the deck planking and that of the topsides, and are caulked similarly.

Modern "constructed decks" are used primarily on [[fiberglass]], composite, and cold-molded hulls. The under structure of beams and carlins is the same as above. The decking itself is usually multiple layers of marine-grade [[plywood]], covered over with layers of fibreglass in a plastic resin such as [[epoxy]] or [[polyester]] overlapped onto the [[#Glossary|topsides]] of the hull.

=== Methods in metal ===

Generally speaking, the method outlined for "constructed decks" is most similar to [[metal]] decks. The deck [[#Glossary|plating]] is laid over metal [[Beam (structure)|beams]] and carlins and tacked temporarily in place. The difficulty in metal construction is avoiding distortion of the plate while [[welding]] due to the high [[heat]] involved in the process. Welds are usually double pass, meaning each seam is welded twice, a time consuming process which may take longer than building the wood deck. But welds result in a waterproof deck which is strong and easily repairable. The deck structure is welded to the hull, making it structurally a single unit.

Because a metal deck, painted to reduce [[corrosion]], can be quite slippery, pick up heat from the [[sun]], and be quite loud to work on, a layer of wood decking or thick non-skid paint is often applied to its surface.

=== Methods in fiberglass ===

The process for building a deck in fiberglass is the same as for building a hull: a female mould is built, a layer of [[#Glossary|gel coat]] is sprayed in, then layers of fiberglass in resin are built up to the required deck thickness (if the deck has a [[#Glossary|core]], the outer skin layers of fiberglass and resin are laid, then the core material, and finally the inner skin layers.) The deck is removed from the mould and usually mechanically fastened to the hull.

Fiberglass decks are quite slick with their mirror-smooth surfaces, so a non-skid texture is often moulded into their surface, or non-skid pads glued down in working areas.

=== Rules of thumb to determine the deck scantlings ===

The thickness of the decking affects how strong the hull is, and is directly related to how thick the skin of the hull itself is, which is of course related to how large the vessel is, the kind of work it is expected to do, and the kind of weather it may reasonably be expected to endure. While a Naval Engineer or Architect may have precise methods of determining what the scantlings should be, traditional builders used previous experiences and simpler rules-of-thumb to determine how thick the deck should be built.

The numbers derived by these formulae gives a rough number for determining the average thickness of materials based on some crude hull measurements. Below the waterline the thickness should be approximately 115% of the result, while upper topsides and decks might be reduced to 85% of the result.

* In wood – For plank thickness in inches, LOA ([[Length overall|Length OverAll]]) and Beam are measured in feet. For plank thickness in mm, LOA and Beam are measured in meters.
**Plank thickness in inches = <math>{\sqrt{LOA}+Beam \over 16}</math>
**Plank thickness in mm = <math>[\sqrt{LOA\cdot 3.28}+(Beam\cdot 3.28)\cdot 1.58]</math>
* In fiberglass – For skin thickness in inches, LWL (Length WaterLine) is in feet. For skin thickness in mm, LWL is in meters.
**Skin thickness (inches) = <math>0.07 + {LWL\over150}</math>
**Skin thickness (mm) = <math>1.8 + {LWL\over1.8}</math>
* In fiberglass sandwich – First determine the skin thickness as single skin, then multiply by modifiers for inner skin, outer skin, and core thicknesses. Cored decks might be modified even thicker, 2.6–2.7, to increase stiffness.
** Inner skin modifier = 0.3
** Outer skin modifier = 0.4
** Core modifier = 2.2
<ref>Gerr, David; ''The Nature of Boats: Insights and esoterica for the nautically obsessed''; International Marine; 1992 International Marine; ISBN 0-87742-289-3.</ref>

== Glossary ==

* athwartships: perpendicular to fore and aft.
* beam: a timber similar in use to a floor joist, which runs from one side of the hull to the other athwartships.
* carlin: similar to a beam, except running in a fore and aft direction.
* caulk: to make water-tight by driving caulking (usually loose cotton fibers) into a seam, followed by a coarser fiber material such as oakum.
* core: in fibreglass construction, a layer between fiberglass skins, made of foam, end grain balsa, or other strengthening material to increase the stiffness of the deck.
* fore and aft: parallel to a line from the stem to the stern.
* gel coat: a heavily pigmented layer of plastic resin.
* [[oakum]]: loosely twisted hemp or [[jute]] or other crude fibre, sometimes treated with creosote or tar before use.
* pay: to pour into or fill up a seam so it is level with the top of the plank.
* plating: sheets of metal, generally simple flat pieces but may be formed into complex curvatures.
* pooping wave: A wave which comes over the stern and onto the deck.
* scantling: the critical dimensions of any element of the ship; so for the skin and deck of the hull it would be the thickness (of the planks, fibreglass layup, hull plating, etc.)
* seam: the space between two planks.
* stem: The timber at the front of the hull.
* stern: back end of the hull.
* topsides: the upper surfaces of the hull from the waterline to the deck.

==Notes==
{{reflist}}

==External links==
* [http://www.hmsvictory.de/web/index.phtml/1131 The history of the deck in old North European ships and languages]
* [http://bestshippingnews.com/shipbuilding-picture-dictionary/equipment-on-forecastle-deck-of-ship/ Equipment on forecastle deck of ship by Picture]

{{Sailing ship elements}}

[[Category:Ship compartments]]
[[Category:Watercraft components]]