Misplaced Pages

#91908

33-425: (Redirected from Point-class ) Point class may refer to Pointclass sets in mathematics Point-class sealift ship Point-class cutter Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Point class . If an internal link led you here, you may wish to change the link to point directly to

66-419: A conservative extension of ZFC. Morse–Kelley set theory admits proper classes as basic objects, like NBG, but also allows quantification over all proper classes in its class existence axioms. This causes MK to be strictly stronger than both NBG and ZFC. In other set theories, such as New Foundations or the theory of semisets , the concept of "proper class" still makes sense (not all classes are sets) but

99-460: A product space to be any finite Cartesian product of these underlying spaces. Then, for example, the pointclass Σ 1 0 {\displaystyle {\boldsymbol {\Sigma }}_{1}^{0}} of all open sets means the collection of all open subsets of one of these product spaces. This approach prevents Σ 1 0 {\displaystyle {\boldsymbol {\Sigma }}_{1}^{0}} from being

132-655: A proper class , while avoiding excessive specificity as to the particular Polish spaces being considered (given that the focus is on the fact that Σ 1 0 {\displaystyle {\boldsymbol {\Sigma }}_{1}^{0}} is the collection of open sets, not on the spaces themselves). The pointclasses in the Borel hierarchy , and in the more complex projective hierarchy , are represented by sub- and super-scripted Greek letters in boldface fonts; for example, Π 1 0 {\displaystyle {\boldsymbol {\Pi }}_{1}^{0}}

165-428: A Polish space (say, an arbitrary real number, or an arbitrary countable sequence of natural numbers). Boldface pointclasses, however, may (and in practice ordinarily do) require that sets in the class be definable relative to some real number, taken as an oracle . In that sense, membership in a boldface pointclass is a definability property, even though it is not absolute definability, but only definability with respect to

198-411: A historical period where classes and sets were not distinguished as they are in modern set-theoretic terminology. Many discussions of "classes" in the 19th century and earlier are really referring to sets, or rather perhaps take place without considering that certain classes can fail to be sets. The collection of all algebraic structures of a given type will usually be a proper class. Examples include

231-401: A lightface Σ {\displaystyle \Sigma } , are no longer arbitrary unions of such neighborhoods, but computable unions of them. That is, a set is lightface Σ 1 0 {\displaystyle \Sigma _{1}^{0}} , also called effectively open , if there is a computable set S of finite sequences of naturals such that the given set

264-510: A possibly undefinable real number. Boldface pointclasses, or at least the ones ordinarily considered, are closed under Wadge reducibility ; that is, given a set in the pointclass, its inverse image under a continuous function (from a product space to the space of which the given set is a subset) is also in the given pointclass. Thus a boldface pointclass is a downward-closed union of Wadge degrees . The Borel and projective hierarchies have analogs in effective descriptive set theory in which

297-400: Is a union of a computable sequence of Π 1 0 {\displaystyle \Pi _{1}^{0}} sets (that is, there is a computable enumeration of indices of Π 1 0 {\displaystyle \Pi _{1}^{0}} sets such that A is the union of these sets). This relationship between lightface sets and their indices is used to extend

330-565: Is at least as large as the classes above it. Proper class In set theory and its applications throughout mathematics , a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. Classes act as a way to have set-like collections while differing from sets so as to avoid paradoxes, especially Russell's paradox (see § Paradoxes ). The precise definition of "class" depends on foundational context. In work on Zermelo–Fraenkel set theory ,

363-450: Is necessary to be able to expand each of the formulas x ∈ A {\displaystyle x\in A} , x = A {\displaystyle x=A} , A ∈ x {\displaystyle A\in x} , and A = x {\displaystyle A=x} into a formula without an occurrence of a class. Semantically, in a metalanguage ,

SECTION 10

#1732773303092

396-445: Is no notion of classes containing classes. Otherwise, one could, for example, define a class of all classes that do not contain themselves, which would lead to a Russell paradox for classes. A conglomerate , on the other hand, can have proper classes as members. ZF set theory does not formalize the notion of classes, so each formula with classes must be reduced syntactically to a formula without classes. For example, one can reduce

429-490: Is the pointclass of all analytic sets . Sets in such pointclasses need be "definable" only up to a point. For example, every singleton set in a Polish space is closed, and thus Π 1 0 {\displaystyle {\boldsymbol {\Pi }}_{1}^{0}} . Therefore, it cannot be that every Π 1 0 {\displaystyle {\boldsymbol {\Pi }}_{1}^{0}} set must be "more definable" than an arbitrary element of

462-495: Is the pointclass of all closed sets , Σ 2 0 {\displaystyle {\boldsymbol {\Sigma }}_{2}^{0}} is the pointclass of all F σ sets, Δ 2 0 {\displaystyle {\boldsymbol {\Delta }}_{2}^{0}} is the collection of all sets that are simultaneously F σ and G δ , and Σ 1 1 {\displaystyle {\boldsymbol {\Sigma }}_{1}^{1}}

495-530: Is the union of the sets { x ∈ω ∣ {\displaystyle \mid } s is an initial segment of x } for s in S . A set is lightface Π 1 0 {\displaystyle \Pi _{1}^{0}} if it is the complement of a Σ 1 0 {\displaystyle \Sigma _{1}^{0}} set. Thus each Σ 1 0 {\displaystyle \Sigma _{1}^{0}} set has at least one index , which describes

528-448: The determinacy of various pointclasses, which in turn implies that sets in those pointclasses (or sometimes larger ones) have regularity properties such as Lebesgue measurability (and indeed universal measurability ), the property of Baire , and the perfect set property . In practice, descriptive set theorists often simplify matters by working in a fixed Polish space such as Baire space or sometimes Cantor space , each of which has

561-497: The "class of all sets" with the set of all predicates equivalent to x = x {\displaystyle x=x} . Because classes do not have any formal status in the theory of ZF, the axioms of ZF do not immediately apply to classes. However, if an inaccessible cardinal κ {\displaystyle \kappa } is assumed, then the sets of smaller rank form a model of ZF (a Grothendieck universe ), and its subsets can be thought of as "classes". In ZF,

594-463: The advantage of being zero dimensional , and indeed homeomorphic to its finite or countable powers , so that considerations of dimensionality never arise. Yiannis Moschovakis provides greater generality by fixing once and for all a collection of underlying Polish spaces, including the set of all naturals, the set of all reals, Baire space, and Cantor space, and otherwise allowing the reader to throw in any desired perfect Polish space. Then he defines

627-438: The class of all groups , the class of all vector spaces , and many others. In category theory , a category whose collection of objects forms a proper class (or whose collection of morphisms forms a proper class) is called a large category . The surreal numbers are a proper class of objects that have the properties of a field . Within set theory, many collections of sets turn out to be proper classes. Examples include

660-418: The class of all sets (the universal class), the class of all ordinal numbers, and the class of all cardinal numbers . One way to prove that a class is proper is to place it in bijection with the class of all ordinal numbers. This method is used, for example, in the proof that there is no free complete lattice on three or more generators . The paradoxes of naive set theory can be explained in terms of

693-443: The class of all sets, are proper classes in many formal systems. In Quine 's set-theoretical writing, the phrase "ultimate class" is often used instead of the phrase "proper class" emphasising that in the systems he considers, certain classes cannot be members, and are thus the final term in any membership chain to which they belong. Outside set theory, the word "class" is sometimes used synonymously with "set". This usage dates from

SECTION 20

#1732773303092

726-416: The classes can be described as equivalence classes of logical formulas : If A {\displaystyle {\mathcal {A}}} is a structure interpreting ZF, then the object language "class-builder expression" { x ∣ ϕ } {\displaystyle \{x\mid \phi \}} is interpreted in A {\displaystyle {\mathcal {A}}} by

759-470: The collection of all the elements from the domain of A {\displaystyle {\mathcal {A}}} on which λ x ϕ {\displaystyle \lambda x\phi } holds; thus, the class can be described as the set of all predicates equivalent to ϕ {\displaystyle \phi } (which includes ϕ {\displaystyle \phi } itself). In particular, one can identify

792-468: The computable function enumerating the basic open sets from which it is composed; in fact it will have infinitely many such indices. Similarly, an index for a Π 1 0 {\displaystyle \Pi _{1}^{0}} set B describes the computable function enumerating the basic open sets in the complement of B . A set A is lightface Σ 2 0 {\displaystyle \Sigma _{2}^{0}} if it

825-419: The concept of a function can also be generalised to classes. A class function is not a function in the usual sense, since it is not a set; it is rather a formula Φ ( x , y ) {\displaystyle \Phi (x,y)} with the property that for any set x {\displaystyle x} there is no more than one set y {\displaystyle y} such that

858-759: The definability property is no longer relativized to an oracle, but is made absolute. For example, if one fixes some collection of basic open neighborhoods (say, in Baire space, the collection of sets of the form { x ∈ω ∣ {\displaystyle \mid } s is an initial segment of x } for each fixed finite sequence s of natural numbers), then the open, or Σ 1 0 {\displaystyle {\boldsymbol {\Sigma }}_{1}^{0}} , sets may be characterized as all (arbitrary) unions of basic open neighborhoods. The analogous Σ 1 0 {\displaystyle \Sigma _{1}^{0}} sets, with

891-479: The formula A = { x ∣ x = x } {\displaystyle A=\{x\mid x=x\}} to ∀ x ( x ∈ A ↔ x = x ) {\displaystyle \forall x(x\in A\leftrightarrow x=x)} . For a class A {\displaystyle A} and a set variable symbol x {\displaystyle x} , it

924-522: The inconsistent tacit assumption that "all classes are sets". With a rigorous foundation, these paradoxes instead suggest proofs that certain classes are proper (i.e., that they are not sets). For example, Russell's paradox suggests a proof that the class of all sets which do not contain themselves is proper, and the Burali-Forti paradox suggests that the class of all ordinal numbers is proper. The paradoxes do not arise with classes because there

957-513: The intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Point_class&oldid=722106115 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Pointclass Pointclasses find application in formulating many important principles and theorems from set theory and real analysis . Strong set-theoretic principles may be stated in terms of

990-469: The lightface Borel hierarchy into the transfinite, via recursive ordinals . This produces the hyperarithmetic hierarchy , which is the lightface analog of the Borel hierarchy. (The finite levels of the hyperarithmetic hierarchy are known as the arithmetical hierarchy .) A similar treatment can be applied to the projective hierarchy. Its lightface analog is known as the analytical hierarchy . Each class

1023-472: The notion of class is informal, whereas other set theories, such as von Neumann–Bernays–Gödel set theory , axiomatize the notion of "proper class", e.g., as entities that are not members of another entity. A class that is not a set (informally in Zermelo–Fraenkel) is called a proper class , and a class that is a set is sometimes called a small class . For instance, the class of all ordinal numbers , and

Point class - Misplaced Pages Continue

1056-533: The pair ( x , y ) {\displaystyle (x,y)} satisfies Φ {\displaystyle \Phi } . For example, the class function mapping each set to its powerset may be expressed as the formula y = P ( x ) {\displaystyle y={\mathcal {P}}(x)} . The fact that the ordered pair ( x , y ) {\displaystyle (x,y)} satisfies Φ {\displaystyle \Phi } may be expressed with

1089-462: The shorthand notation Φ ( x ) = y {\displaystyle \Phi (x)=y} . Another approach is taken by the von Neumann–Bernays–Gödel axioms (NBG); classes are the basic objects in this theory, and a set is then defined to be a class that is an element of some other class. However, the class existence axioms of NBG are restricted so that they only quantify over sets, rather than over all classes. This causes NBG to be

#91908