Everything about Class Set Theory totally explained
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. The precise definition of "class" depends on foundational context. In work on
ZF set theory, the notion of class is informal, whereas other set theories, such as
NBG set theory, axiomatize the notion of "class".
Every set is a class, no matter which foundation is chosen. A class that isn't a set is (informally) called a
proper class, and a class that's a set is sometimes called a
small class. For instance, the class of all
ordinal numbers, and the class of all sets, are proper classes in many formal systems.
Various important concepts in mathematics are commonly described with classes. Examples include large
categories or the class-field of
surreal numbers.
In
ZF set theory, classes exist only in the
metalanguage, as equivalence classes of logical formulas. The axioms of ZF don't apply to classes. However, if an
inaccessible cardinal κ is assumed, then the sets of smaller cardinality form a model of ZF (a
Grothendieck universe), and all the sets of larger cardinality can be thought of as "classes".
Another approach is taken by the
von Neumann-Bernays-Gödel axioms; classes are the basic objects in this theory, and a set is then defined to be a class that's an element of some other class. In other, less standard 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 the criterion of sethood isn't closed under subsets. For example, any set theory with a universal set has proper classes which are subclasses of sets.
The
paradoxes of naive set theory can be explained in terms of the inconsistent assumption that "all classes are sets". With a rigorous foundation, these paradoxes instead suggest
proofs that certain classes are proper. For example,
Russell's paradox suggests a proof that the class of all sets which don't contain themselves is proper, and the
Burali-Forti paradox suggests that the class of all
ordinal numbers is proper. One way to prove that a class is proper is to place it in bijection with the class of ordinals; see, for instance, the proof that there's no
free complete lattice.
The word "class" is sometimes used synonymously with "set". This usage dates from a historical period where classes and sets were not distinguished as they're in modern terminology. Many discussions of "classes" in the
19th century and earlier are really referring to sets, or perhaps to a more ambiguous concept.
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