Class (set theory)

In mathematics , class logic and set theory, a class is a summary of any objects, defined by a logical property that all objects of the class fulfill. The concept of sets is to be distinguished from the concept of class. Not all classes are automatically sets because sets have to meet additional conditions. However, sets are always classes and are therefore notated in class notation in practice.

About history

In 19th century mathematics, the terms “class” and “ quantity ” were largely used synonymously and were insufficiently defined, so that contradicting interpretations were possible. In the 20th century, in the course of the axiomatization of set theory, they were separated and gradually made more precise. Since then, the term “class” has often been used more broadly than the term “set”.

Classes are not restricted in their formation or definition. Often, however, they may only be used to a limited extent, so that the contradictions of naive set theory do not arise. For example, not every class can be a member of sets. Only the improper handling of classes is therefore problematic and creates contradictions.

Definitions

If there is any logically correct statement with the variable , then the totality of all objects that satisfy the statement is referred to as a class and noted as or . The definition also applies to variables that do not appear in the statement ; and are bound variables here . ${\ displaystyle A (x)}$ ${\ displaystyle x}$ ${\ displaystyle x}$ ${\ displaystyle A (x)}$ ${\ displaystyle \ {x \ mid A (x) \}}$ ${\ displaystyle \ {x: A (x) \}}$ ${\ displaystyle \ {y \ mid A (y) \} = \ {x \ mid A (x) \}}$ ${\ displaystyle y}$ ${\ displaystyle A (x)}$ ${\ displaystyle x}$ ${\ displaystyle y}$

Classes in this representation and notation are used everywhere in mathematics practice today, regardless of which axiomatic basis is assumed. For its application it is therefore not decisive whether the Zermelo-Fraenkel set theory (ZF) or the Neumann-Bernays-Gödel set theory (NBG) or another system of axioms is used. In ZF and NBG, however, classes are not official terms , but are only used for practical purposes; Strictly speaking, there is an unofficial class notation that does not strictly belong to the formal language. Only through additional axiom schemes are they correctly incorporated into the logical language, in ZF through the following three principles: ${\ displaystyle \ {x \ mid A (x) \}}$

(1) The abstraction principle covers the class property mentioned in the definition:

{\ displaystyle \ forall y \ colon (y \ in \ {x \ mid A (x) \} \ iff A (y))}

(2) The extensionality principle describes the equality of classes through the correspondence of their elements:

{\ displaystyle A = B \ iff \ forall x \ colon (x \ in A \ iff x \ in B)}

NB: This extensionality principle has free variables for classes (capital letters). It implies the quantified axiom of extensionality for sets in ZF.

(3) The principle of comprehension defines the existence of a class as an element:

{\ displaystyle \ {x \ mid A (x) \} \ in B \ iff \ exists y \ colon (y = \ {x \ mid A (x) \} \ land y \ in B)}

With these three principles, cumbersome formulas of the predicate logic ZF language can be translated into convenient and more understandable formulas with classes. They can be understood as additional axioms for so-called virtual classes (see below). They also apply when using class terms (see below) in the context of class logic ; there a class term says nothing about the existence of a class! The class logic is therefore only a syntactically rich logical framework, which allows a more convenient, optimized representation and allows any classes to be used in any context without the risk of contradiction. Class variables are free variables here; In contrast, only elements can be used in bound variables, especially all sets that must meet the criterion in the comprehension principle.

Classes can be linked, such as quantities, namely with the operators of a same operators Boolean Association and with the element predicate . In the same way , the definitions customary in set theory can also be transferred to classes, such as the partial predicate , the power , the union , the average , the Cartesian product or ordered pairs . All basic theorems then also apply; some special theorems of set theory, which presuppose certain set formations (existing classes), are not valid because sets are defined differently in different set theories. But it always applies that a lot is a class. The converse does not apply, however, because due to the contradictions of naive set theory not all classes are also sets. ${\ displaystyle \ cap}$ ${\ displaystyle \ cup}$ ${\ displaystyle \ in}$ ${\ displaystyle \ subseteq}$ ${\ displaystyle {\ mathcal {P}} (A)}$ ${\ displaystyle \ bigcup A}$ ${\ displaystyle \ bigcap A}$ ${\ displaystyle A \ times B}$ ${\ displaystyle (A, B)}$

Real classes

Classes that no amounts are usually called real or actual classes . This means that real classes do not fulfill certain axioms of set theory, whereby mostly the axioms of Zermelo-Fraenkel set theory (ZF) are meant, but in principle other axiomatic set theories are also possible. The real classes include, in particular, all classes that cannot be an element of another class or set, since the set can always be formed from the set . ${\ displaystyle x}$ ${\ displaystyle \ {x \}}$

Examples of real classes:

The class of all objects, the so-called universal class : . In set theory this is the class of all sets.

{\ displaystyle \ {x \ mid x = x \}}

The class of all sets that do not contain themselves as an element, the so-called Russell's class : . In the Zermelo-Fraenkel set theory (ZF) this is equal to the all-class.

{\ displaystyle \ {x \ mid x \ notin x \}}

The class of all one-element sets.

The class of all ordinals .

The class of all cardinal numbers .

The class of all objects of a certain category is often a real class, for example the class of all groups or the class of all vector spaces over a body. From the example of the class of all one-element sets it follows that the class of all trivial groups is already a real class. But since a group of this order or a vector space of this dimension also exists for every cardinal number, there is also no equivalent sub-category whose objects form a set. In contrast, the full subcategory of vector spaces for natural ones is equivalent to the category of all finite-dimensional vector spaces.

{\ displaystyle K ^ {n}}

{\ displaystyle n}

The class of surreal numbers . It has all the properties of a body, except the property of being a lot.

Quine individuals with . They violate the axiom of foundations in set theory . ${\ displaystyle \ {x \} = x}$

Informally, one can say that a class is real when it is “too big” to be a lot; therefore one speaks unofficially of “vast amounts” in allusion to the colloquial meaning of an unmanageable amount. For example, the class of all integers is a set - infinitely large, but still manageable; the class of all groups, on the other hand, as well as the class of all sets, are “too big” and therefore real classes. The converse that real classes are always too large classes does not necessarily apply, because there are also small real classes in certain set theories, as the last example shows.

Real classes are not subject to the set axioms. For example, the power of the universal class violates Cantor's second diagonal argument for power sets ; Cantor used this Cantorian antinomy as an indirect proof that the universal class is not a set, but a real class. Other paradoxes of naive set theory also indirectly prove that a certain class is real: The Burali-Forti paradox is proof of the authenticity of the class of all ordinals and Russell's antinomy is proof of the authenticity of the Russell class.

Virtual classes

Virtual classes were introduced by Quine as class formulas, which are not independent terms, but rather partial formulas in fixed logical contexts. He used this technique because ZF set theory is based on predicate logic with element predicate and, strictly speaking, has no class terms of the form ; these cannot be correctly defined there because only predicate logic statements are available as formulas. Three established contexts for virtual classes are the above principles (1) (2) (3). They extend the ZF set language in such a way that all sets can be noted as classes; You can also virtually note all real classes, even if they are not existing objects in ZF. ${\ displaystyle \ {x \ mid A (x) \}}$ ${\ displaystyle \ {x \ mid A (x) \}}$

Class terms

If one chooses a class logic as the basis instead of a predicate logic , then any class becomes a correct, fully-fledged term . This is possible, for example, in Oberschelp set theory , which is a further development of Quine set theory to form a ZFC class logic. You can also choose this basis for NBG. Only such class-logical versions of set theory offer the optimum convenience for precise set language that does justice to mathematical practice in every respect. The above-mentioned principles (1) (2) (3) apply here, in particular the quantified abstraction principle (1). However, the naive, more general and unquantified abstraction principle of Frege does not apply , since it is contradictory because of the free variables and, by inserting Russell's class, creates Russell's antinomy . ${\ displaystyle \ {x \ mid A (x) \}}$ ${\ displaystyle y \ in \ {x \ mid A (x) \} \ iff A (y)}$ ${\ displaystyle y}$

literature

Arnold Oberschelp: General set theory. BI-Wissenschafts-Verlag, Mannheim et al. 1994, ISBN 3-411-17271-1 .
Willard Van Orman Quine: Set theory and its logic. (= Logic and Fundamentals of Mathematics. Vol. 10). Vieweg, Braunschweig 1973, ISBN 3-528-08294-1 .

Individual evidence and explanations

^ Arnold Oberschelp, p. 262, 41.7.
↑ Ackermann set theory also has an equivalent axiom of extensionality with free variables for any classes .
^ Arnold Oberschelp, pp. 38–41.
^ Arnold Oberschelp, p. 230.
↑ University of Heidelberg: Mathematical Logic WS 2013/14 Chapter 5: Set theory
^ Willard Van Orman Quine, p. 24.
↑ Willard Van Orman Quine, p. 12.
↑ Gottlob Frege : Fundamental laws of arithmetic. Derived conceptually. Volume 1. Pohle, Jena 1893, p. 52.

[1] Arnold Oberschelp, p. 262, 41.7.

[2] Ackermann set theory also has an equivalent axiom of extensionality with free variables for any classes .

[3] Arnold Oberschelp, pp. 38–41.

[4] Arnold Oberschelp, p. 230.

[5] University of Heidelberg: Mathematical Logic WS 2013/14 Chapter 5: Set theory

[6] Willard Van Orman Quine, p. 24.

[7] Willard Van Orman Quine, p. 12.

[8] Gottlob Frege : Fundamental laws of arithmetic. Derived conceptually. Volume 1. Pohle, Jena 1893, p. 52.