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Inset theory, anordinal number, orordinal, is a generalization ofordinal numerals (first, second,nth, etc.) aimed to extendenumeration toinfinite sets.[1]
A finite set can be enumerated by successively labeling each element with the leastnatural number that has not been previously used. To extend this process to variousinfinite sets, ordinal numbers are defined more generally usinglinearly orderedgreek lettervariables that include the natural numbers and have the property that every set of ordinals has aleast or "smallest" element (this is needed for giving a meaning to "the least unused element").[2] This more general definition allows us to define an ordinal number (omega) to be the least element that is greater than every natural number, along with ordinal numbers,, etc., which are even greater than.
A linear order such that every non-empty subset has a least element is called awell-order. Theaxiom of choice implies that every set can be well-ordered, and given two well-ordered sets, one isisomorphic to aninitial segment of the other. So ordinal numbers exist and are essentially unique.
Ordinal numbers are distinct fromcardinal numbers, which measure the size of sets. Although the distinction between ordinals and cardinals is not always apparent on finite sets (one can go from one to the other just by counting labels), they are very different in the infinite case, where different infinite ordinals can correspond to sets having the same cardinal. Like other kinds of numbers, ordinals can beadded, multiplied, and exponentiated, although none of these operations arecommutative.
Ordinals were introduced byGeorg Cantor in 1883[3] in order to accommodate infinite sequences and classifyderived sets, which he had previously introduced in 1872 while studying the uniqueness oftrigonometric series.[4]
Anatural number (which, in this context, includes the number0) can be used for two purposes: to describe thesize of aset, or to describe theposition of an element in a sequence. When restricted to finite sets, these two concepts coincide, since alllinear orders of a finite set areisomorphic.
When dealing with infinite sets, however, one has to distinguish between the notion of size, which leads tocardinal numbers, and the notion of position, which leads to the ordinal numbers described here. This is because while any set has only one size (itscardinality), there are many nonisomorphicwell-orderings of any infinite set, as explained below.
Whereas the notion of cardinal number is associated with a set with no particular structure on it, the ordinals are intimately linked with the special kind of sets that are calledwell-ordered. A well-ordered set is atotally ordered set (anordered set such that, given two distinct elements, one is less than the other) in which every non-empty subset has a least element. Equivalently, assuming theaxiom of dependent choice, it is a totally ordered set without any infinite decreasing sequence — though there may be infinite increasing sequences. Ordinals may be used to label the elements of any given well-ordered set (the smallest element being labelled 0, the one after that 1, the next one 2, "and so on"), and to measure the "length" of the whole set by the least ordinal that is not a label for an element of the set. This "length" is called theorder type of the set.
Any ordinal is defined by the set of ordinals that precede it. In fact, the most common definition of ordinalsidentifies each ordinalas the set of ordinals that precede it. For example, the ordinal 42 is generally identified as the set {0, 1, 2, ..., 41}. Conversely, any setS of ordinals that isdownward closed — meaning that for any ordinal α inS and any ordinal β < α, β is also inS — is (or can be identified with) an ordinal.
This definition of ordinals in terms of sets allows for infinite ordinals. The smallest infinite ordinal is, which can be identified with the set of natural numbers (so that the ordinal associated with every natural number precedes). Indeed, the set of natural numbers is well-ordered—as is any set of ordinals—and since it is downward closed, it can be identified with the ordinal associated with it.
Perhaps a clearer intuition of ordinals can be formed by examining a first few of them: as mentioned above, they start with the natural numbers, 0, 1, 2, 3, 4, 5, ... Afterall natural numbers comes the first infinite ordinal, ω, and after that come ω+1, ω+2, ω+3, and so on. (Exactly what addition means will be defined later on: just consider them as names.) After all of these come ω·2 (which is ω+ω), ω·2+1, ω·2+2, and so on, then ω·3, and then later on ω·4. Now the set of ordinals formed in this way (the ω·m+n, wherem andn are natural numbers) must itself have an ordinal associated with it: and that is ω2. Further on, there will be ω3, then ω4, and so on, and ωω, then ωωω, then later ωωωω, and even later ε0 (epsilon nought) (to give a few examples of relatively small—countable—ordinals). This can be continued indefinitely (as every time one says "and so on" when enumerating ordinals, it defines a larger ordinal). The smallestuncountable ordinal is the set of all countable ordinals, expressed asω1 or.[5]
In awell-ordered set, every non-empty subset contains a distinct smallest element. Given theaxiom of dependent choice, this is equivalent to saying that the set istotally ordered and there is no infinite decreasing sequence (the latter being easier to visualize). In practice, the importance of well-ordering is justified by the possibility of applyingtransfinite induction, which says, essentially, that any property that passes on from the predecessors of an element to that element itself must be true of all elements (of the given well-ordered set). If the states of a computation (computer program or game) can be well-ordered—in such a way that each step is followed by a "lower" step—then the computation will terminate.
It is inappropriate to distinguish between two well-ordered sets if they only differ in the "labeling of their elements", or more formally: if the elements of the first set can be paired off with the elements of the second set such that if one element is smaller than another in the first set, then the partner of the first element is smaller than the partner of the second element in the second set, and vice versa. Such a one-to-one correspondence is called anorder isomorphism, and the two well-ordered sets are said to be order-isomorphic orsimilar (with the understanding that this is anequivalence relation).
Formally, if apartial order ≤ is defined on the setS, and a partial order ≤' is defined on the setS', then theposets (S,≤) and (S',≤') areorder isomorphic if there is abijectionf that preserves the ordering. That is,f(a) ≤'f(b) if and only ifa ≤b. Provided there exists an order isomorphism between two well-ordered sets, the order isomorphism is unique: this makes it quite justifiable to consider the two sets as essentially identical, and to seek a"canonical" representative of the isomorphism type (class). This is exactly what the ordinals provide, and it also provides a canonical labeling of the elements of any well-ordered set. Everywell-ordered set (S,<) is order-isomorphic to the set of ordinals less than one specific ordinal number under their natural ordering. This canonical set is theorder type of (S,<).
Essentially, an ordinal is intended to be defined as anisomorphism class of well-ordered sets: that is, as anequivalence class for theequivalence relation of "being order-isomorphic". There is a technical difficulty involved, however, in the fact that the equivalence class is too large to be a set in the usualZermelo–Fraenkel (ZF) formalization of set theory. But this is not a serious difficulty. The ordinal can be said to be theorder type of any set in the class.
The original definition of ordinal numbers, found for example in thePrincipia Mathematica, defines the order type of a well-ordering as the set of all well-orderings similar (order-isomorphic) to that well-ordering: in other words, an ordinal number is genuinely an equivalence class of well-ordered sets. This definition must be abandoned inZF and related systems ofaxiomatic set theory because these equivalence classes are too large to form a set. However, this definition still can be used intype theory and in Quine's axiomatic set theoryNew Foundations and related systems (where it affords a rather surprising alternative solution to theBurali-Forti paradox of the largest ordinal).
| 0 | = | {} | = | ∅ |
|---|---|---|---|---|
| 1 | = | {0} | = | {∅} |
| 2 | = | {0,1} | = | {∅,{∅}} |
| 3 | = | {0,1,2} | = | {∅,{∅},{∅,{∅}}} |
| 4 | = | {0,1,2,3} | = | {∅,{∅},{∅,{∅}},{∅,{∅},{∅,{∅}}}} |
Rather than defining an ordinal as anequivalence class of well-ordered sets, it will be defined as a particular well-ordered set that (canonically) represents the class. Thus, an ordinal number will be a well-ordered set; and every well-ordered set will be order-isomorphic to exactly one ordinal number.
For each well-ordered setT, defines anorder isomorphism betweenT and the set of all subsets ofT having the form ordered by inclusion. This motivates the standard definition, suggested byJohn von Neumann at the age of 19, now called definition ofvon Neumann ordinals: "each ordinal is the well-ordered set of all smaller ordinals". In symbols,.[6][7] Formally:
The natural numbers are thus ordinals by this definition. For instance, 2 is an element of4 = {0, 1, 2, 3}, and 2 is equal to{0, 1} and so it is a subset of{0, 1, 2, 3}.
It can be shown bytransfinite induction that every well-ordered set is order-isomorphic to exactly one of these ordinals, that is, there is an order preservingbijective function between them.
Furthermore, the elements of every ordinal are ordinals themselves. Given two ordinalsS andT,S is an element ofT if and only ifS is aproper subset ofT. Moreover, eitherS is an element ofT, orT is an element ofS, or they are equal. So every set of ordinals istotally ordered. Further, every set of ordinals is well-ordered. This generalizes the fact that every set of natural numbers is well-ordered.
Consequently, every ordinalS is a set having as elements precisely the ordinals smaller thanS. For example, every set of ordinals has asupremum, the ordinal obtained by taking the union of all the ordinals in the set. This union exists regardless of the set's size, by theaxiom of union.
The class of all ordinals is not a set. If it were a set, one could show that it was an ordinal and thus a member of itself, which would contradict itsstrict ordering by membership. This is theBurali-Forti paradox. The class of all ordinals is variously called "Ord", "ON", or "∞".
An ordinal isfinite if and only if the opposite order is also well-ordered, which is the case if and only if each of its non-empty subsets has agreatest element.
There are other modern formulations of the definition of ordinal. For example, assuming theaxiom of regularity, the following are equivalent for a setx:
These definitions cannot be used innon-well-founded set theories. In set theories withurelements, one has to further make sure that the definition excludes urelements from appearing in ordinals.
If α is any ordinal andX is a set, an α-indexed sequence of elements ofX is a function from α toX. This concept, atransfinite sequence (if α is infinite) orordinal-indexed sequence, is a generalization of the concept of asequence. An ordinary sequence corresponds to the case α = ω, while a finite α corresponds to atuple, a.k.a.string.
Transfinite induction holds in anywell-ordered set, but it is so important in relation to ordinals that it is worth restating here.
That is, ifP(α) is true wheneverP(β) is true for allβ < α, thenP(α) is true forall α. Or, more practically: in order to prove a propertyP for all ordinals α, one can assume that it is already known for all smallerβ < α.
Transfinite induction can be used not only to prove things, but also to define them. Such a definition is normally said to be bytransfinite recursion – the proof that the result is well-defined uses transfinite induction. LetF denote a (class) functionF to be defined on the ordinals. The idea now is that, in definingF(α) for an unspecified ordinal α, one may assume thatF(β) is already defined for allβ < α and thus give a formula forF(α) in terms of theseF(β). It then follows by transfinite induction that there is one and only one function satisfying the recursion formula up to and including α.
Here is an example of definition by transfinite recursion on the ordinals (more will be given later): define functionF by lettingF(α) be the smallest ordinal not in the set{F(β) | β < α}, that is, the set consisting of allF(β) forβ < α. This definition assumes theF(β) known in the very process of definingF; this apparent vicious circle is exactly what definition by transfinite recursion permits. In fact,F(0) makes sense since there is no ordinalβ < 0, and the set{F(β) | β < 0} is empty. SoF(0) is equal to 0 (the smallest ordinal of all). Now thatF(0) is known, the definition applied toF(1) makes sense (it is the smallest ordinal not in the singleton set{F(0)} = {0}), and so on (theand so on is exactly transfinite induction). It turns out that this example is not very exciting, since provablyF(α) = α for all ordinals α, which can be shown, precisely, by transfinite induction.
Any nonzero ordinal has the minimum element, zero. It may or may not have a maximum element. For example, 42 has maximum 41 and ω+6 has maximum ω+5. On the other hand, ω does not have a maximum since there is no largest natural number. If an ordinal has a maximum α, then it is the next ordinal after α, and it is called asuccessor ordinal, namely the successor of α, written α+1. In the von Neumann definition of ordinals, the successor of α is since its elements are those of α and α itself.[6]
A nonzero ordinal that isnot a successor is called alimit ordinal. One justification for this term is that a limit ordinal is thelimit in a topological sense of all smaller ordinals (under theorder topology).
When is an ordinal-indexed sequence, indexed by a limit and the sequence isincreasing, i.e. whenever itslimit is defined as the least upper bound of the set that is, the smallest ordinal (it always exists) greater than any term of the sequence. In this sense, a limit ordinal is the limit of all smaller ordinals (indexed by itself). Put more directly, it is the supremum of the set of smaller ordinals.
Another way of defining a limit ordinal is to say that α is a limit ordinal if and only if:
So in the following sequence:
ω is a limit ordinal because for any smaller ordinal (in this example, a natural number) there is another ordinal (natural number) larger than it, but still less than ω.
Thus, every ordinal is either zero, or a successor (of a well-defined predecessor), or a limit. This distinction is important, because many definitions by transfinite recursion rely upon it. Very often, when defining a functionF by transfinite recursion on all ordinals, one definesF(0), andF(α+1) assumingF(α) is defined, and then, for limit ordinals δ one definesF(δ) as the limit of theF(β) for all β<δ (either in the sense of ordinal limits, as previously explained, or for some other notion of limit ifF does not take ordinal values). Thus, the interesting step in the definition is the successor step, not the limit ordinals. Such functions (especially forF nondecreasing and taking ordinal values) are called continuous. Ordinal addition, multiplication and exponentiation are continuous as functions of their second argument (but can be defined non-recursively).
Any well-ordered set is similar (order-isomorphic) to a unique ordinal number; in other words, its elements can be indexed in increasing fashion by the ordinals less than. This applies, in particular, to any set of ordinals: any set of ordinals is naturally indexed by the ordinals less than some. The same holds, with a slight modification, forclasses of ordinals (a collection of ordinals, possibly too large to form a set, defined by some property): any class of ordinals can be indexed by ordinals (and, when the class is unbounded in the class of all ordinals, this puts it in class-bijection with the class of all ordinals). So the-th element in the class (with the convention that the "0-th" is the smallest, the "1-st" is the next smallest, and so on) can be freely spoken of. Formally, the definition is by transfinite induction: the-th element of the class is defined (provided it has already been defined for all), as the smallest element greater than the-th element for all.
This could be applied, for example, to the class of limit ordinals: the-th ordinal, which is either a limit or zero is (seeordinal arithmetic for the definition of multiplication of ordinals). Similarly, one can consideradditively indecomposable ordinals (meaning a nonzero ordinal that is not the sum of two strictly smaller ordinals): the-th additively indecomposable ordinal is indexed as. The technique of indexing classes of ordinals is often useful in the context of fixed points: for example, the-th ordinal such that is written. These are called the "epsilon numbers".
A class of ordinals is said to beunbounded, orcofinal, when given any ordinal, there is a in such that (then the class must be a proper class, i.e., it cannot be a set). It is said to beclosed when the limit of a sequence of ordinals in the class is again in the class: or, equivalently, when the indexing (class-)function is continuous in the sense that, for a limit ordinal, (the-th ordinal in the class) is the limit of all for; this is also the same as being closed, in thetopological sense, for theorder topology (to avoid talking of topology on proper classes, one can demand that the intersection of the class with any given ordinal is closed for the order topology on that ordinal, this is again equivalent).
Of particular importance are those classes of ordinals that areclosed and unbounded, sometimes calledclubs. For example, the class of all limit ordinals is closed and unbounded: this translates the fact that there is always a limit ordinal greater than a given ordinal, and that a limit of limit ordinals is a limit ordinal (a fortunate fact if the terminology is to make any sense at all!). The class of additively indecomposable ordinals, or the class of ordinals, or the class ofcardinals, are all closed unbounded; the set ofregular cardinals, however, is unbounded but not closed, and any finite set of ordinals is closed but not unbounded.
A class is stationary if it has a nonempty intersection with every closed unbounded class. All superclasses of closed unbounded classes are stationary, and stationary classes are unbounded, but there are stationary classes that are not closed and stationary classes that have no closed unbounded subclass (such as the class of all limit ordinals with countable cofinality). Since the intersection of two closed unbounded classes is closed and unbounded, the intersection of a stationary class and a closed unbounded class is stationary. But the intersection of two stationary classes may be empty, e.g. the class of ordinals with cofinality ω with the class of ordinals with uncountable cofinality.
Rather than formulating these definitions for (proper) classes of ordinals, one can formulate them for sets of ordinals below a given ordinal: A subset of a limit ordinal is said to be unbounded (or cofinal) under provided any ordinal less than is less than some ordinal in the set. More generally, one can call a subset of any ordinal cofinal in provided every ordinal less than is less thanor equal to some ordinal in the set. The subset is said to be closed under provided it is closed for the order topologyin, i.e. a limit of ordinals in the set is either in the set or equal to itself.
There are three usual operations on ordinals: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an explicit well-ordered set that represents the operation or by using transfinite recursion. TheCantor normal form provides a standardized way of writing ordinals. It uniquely represents each ordinal as a finite sum of ordinal powers of ω. However, this cannot form the basis of a universal ordinal notation due to such self-referential representations as ε0 = ωε0.
Ordinals are a subclass of the class ofsurreal numbers, and the so-called "natural" arithmetical operations for surreal numbers are an alternative way to combine ordinals arithmetically. They retain commutativity at the expense of continuity.
Interpreted asnimbers, a game-theoretic variant of numbers, ordinals can also be combined via nimber arithmetic operations. These operations are commutative but the restriction to natural numbers is generally not the same as ordinary addition of natural numbers.
Each ordinal associates with onecardinal, its cardinality. If there is a bijection between two ordinals (e.g.ω = 1 + ω andω + 1 > ω), then they associate with the same cardinal. Any well-ordered set having an ordinal as its order-type has the same cardinality as that ordinal. The least ordinal associated with a given cardinal is called theinitial ordinal of that cardinal. Every finite ordinal (natural number) is initial, and no other ordinal associates with its cardinal. But most infinite ordinals are not initial, as many infinite ordinals associate with the same cardinal. Theaxiom of choice is equivalent to the statement that every set can be well-ordered, i.e. that every cardinal has an initial ordinal. In theories with the axiom of choice, the cardinal number of any set has an initial ordinal, and one may employ theVon Neumann cardinal assignment as the cardinal's representation. (However, we must then be careful to distinguish between cardinal arithmetic and ordinal arithmetic.) In set theories without the axiom of choice, a cardinal may be represented by the set of sets with that cardinality having minimal rank (seeScott's trick).
One issue with Scott's trick is that it identifies the cardinal number with, which in some formulations is the ordinal number. It may be clearer to apply Von Neumann cardinal assignment to finite cases and to use Scott's trick for sets which are infinite or do not admit well orderings. Note that cardinal and ordinal arithmetic agree for finite numbers.
The α-th infinite initial ordinal is written, it is always a limit ordinal. Its cardinality is written. For example, the cardinality of ω0 = ω is, which is also the cardinality of ω2 or ε0 (all are countable ordinals). So ω can be identified with, except that the notation is used when writing cardinals, and ω when writing ordinals (this is important since, for example, = whereas). Also, is the smallest uncountable ordinal (to see that it exists, consider the set of equivalence classes of well-orderings of the natural numbers: each such well-ordering defines a countable ordinal, and is the order type of that set), is the smallest ordinal whose cardinality is greater than, and so on, and is the limit of the for natural numbersn (any limit of cardinals is a cardinal, so this limit is indeed the first cardinal after all the).
Thecofinality of an ordinal is the smallest ordinal that is the order type of acofinal subset of. Notice that a number of authors define cofinality or use it only for limit ordinals. The cofinality of a set of ordinals or any other well-ordered set is the cofinality of the order type of that set.
Thus for a limit ordinal, there exists a-indexed strictly increasing sequence with limit. For example, the cofinality of ω2 is ω, because the sequence ω·m (wherem ranges over the natural numbers) tends to ω2; but, more generally, any countable limit ordinal has cofinality ω. An uncountable limit ordinal may have either cofinality ω as does or an uncountable cofinality.
The cofinality of 0 is 0. And the cofinality of any successor ordinal is 1. The cofinality of any limit ordinal is at least.
An ordinal that is equal to its cofinality is called regular and it is always an initial ordinal. Any limit of regular ordinals is a limit of initial ordinals and thus is also initial even if it is not regular, which it usually is not. If the Axiom of Choice, then is regular for each α. In this case, the ordinals 0, 1,,, and are regular, whereas 2, 3,, and ωω·2 are initial ordinals that are not regular.
The cofinality of any ordinalα is a regular ordinal, i.e. the cofinality of the cofinality ofα is the same as the cofinality ofα. So the cofinality operation isidempotent.
As mentioned above (seeCantor normal form), the ordinal ε0 is the smallest satisfying the equation, so it is the limit of the sequence 0, 1,,,, etc. Many ordinals can be defined in such a manner as fixed points of certain ordinal functions (the-th ordinal such that is called, then one could go on trying to find the-th ordinal such that, "and so on", but all the subtlety lies in the "and so on"). One could try to do this systematically, but no matter what system is used to define and construct ordinals, there is always an ordinal that lies just above all the ordinals constructed by the system. Perhaps the most important ordinal that limits a system of construction in this manner is theChurch–Kleene ordinal, (despite the in the name, this ordinal is countable), which is the smallest ordinal that cannot in any way be represented by acomputable function (this can be made rigorous, of course). Considerably large ordinals can be defined below, however, which measure the "proof-theoretic strength" of certainformal systems (for example, measures the strength ofPeano arithmetic). Large countable ordinals such as countableadmissible ordinals can also be defined above the Church-Kleene ordinal, which are of interest in various parts of logic.[citation needed]
Any ordinal number can be made into atopological space by endowing it with theorder topology; this topology isdiscrete if and only if it is less than or equal to ω. A subset of ω + 1 is open in the order topology if and only if either it iscofinite or it does not contain ω as an element.
See theTopology and ordinals section of the "Order topology" article.
The transfinite ordinal numbers, which first appeared in 1883,[8] originated in Cantor's work withderived sets. IfP is a set of real numbers, the derived setP′ is the set oflimit points ofP. In 1872, Cantor generated the setsP(n) by applying the derived set operationn times toP. In 1880, he pointed out that these sets form the sequenceP'⊇ ··· ⊇P(n) ⊇P(n + 1) ⊇ ···, and he continued the derivation process by definingP(∞) as the intersection of these sets. Then he iterated the derived set operation and intersections to extend his sequence of sets into the infinite:P(∞) ⊇P(∞ + 1) ⊇P(∞ + 2) ⊇ ··· ⊇P(2∞) ⊇ ··· ⊇P(∞2) ⊇ ···.[9] The superscripts containing ∞ are just indices defined by the derivation process.[10]
Cantor used these sets in the theorems:
These theorems are proved by partitioningP′ intopairwise disjoint sets:P′ = (P′\P(2)) ∪ (P(2) \P(3)) ∪ ··· ∪ (P(∞) \P(∞ + 1)) ∪ ··· ∪P(α). Forβ <α: sinceP(β + 1) contains the limit points ofP(β), the setsP(β) \P(β + 1) have no limit points. Hence, they arediscrete sets, so they are countable. Proof of first theorem: IfP(α) = ∅ for some indexα, thenP′ is the countable union of countable sets. Therefore,P′ is countable.[11]
The second theorem requires proving the existence of anα such thatP(α) = ∅. To prove this, Cantor considered the set of allα having countably many predecessors. To define this set, he defined the transfinite ordinal numbers and transformed the infinite indices into ordinals by replacing ∞ withω, the first transfinite ordinal number. Cantor called the set of finite ordinals the firstnumber class. The second number class is the set of ordinals whose predecessors form a countably infinite set. The set of allα having countably many predecessors—that is, the set of countable ordinals—is the union of these two number classes. Cantor proved that the cardinality of the second number class is the first uncountable cardinality.[12]
Cantor's second theorem becomes: IfP′ is countable, then there is a countable ordinalα such thatP(α) = ∅. Its proof usesproof by contradiction. LetP′ be countable, and assume there is no such α. This assumption produces two cases.
In both cases,P′ is uncountable, which contradictsP′ being countable. Therefore, there is a countable ordinalα such thatP(α) = ∅. Cantor's work with derived sets and ordinal numbers led to theCantor-Bendixson theorem.[14]
Using successors, limits, and cardinality, Cantor generated an unbounded sequence of ordinal numbers and number classes.[15] The(α + 1)-th number class is the set of ordinals whose predecessors form a set of the same cardinality as theα-th number class. The cardinality of the(α + 1)-th number class is the cardinality immediately following that of theα-th number class.[16] For a limit ordinalα, theα-th number class is the union of theβ-th number classes forβ <α.[17] Its cardinality is the limit of the cardinalities of these number classes.
Ifn is finite, then-th number class has cardinality. Ifα ≥ω, theα-th number class has cardinality.[18] Therefore, the cardinalities of the number classes correspond one-to-one with thealeph numbers. Also, theα-th number class consists of ordinals different from those in the preceding number classes if and only ifα is a non-limit ordinal. Therefore, the non-limit number classes partition the ordinals into pairwise disjoint sets.
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