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Set vs Class (Set Theory)

14 Sep 2016

Intuition and Motivation

The idea behind what we call class is after we have gone to infinity and beyond that. Let us say there are sets; later, they grow bigger and bigger and become infinite. Then, it comes to a question if the collection of what we have accumulated so far is still a set?.

In Naive Set Theory, one can freely collect all objects which satisfy certain properties into a set. Hence, this can easily lead to a paradox. Famous ones are

  • Russell’s paradox
  • Burali-Forti paradox

Class

A class is a collection of sets (or mathematical objects) that can be unambiguously defined by a property that all its members share. Its precise definition depends on a foundational context. For example, Zermelo-Fraenkel set theory defines the notion informally whereas Von Neumann-Bernays-Godel set theory define it as entities that are not members of another entity.

A class that is not a set is called a proper class. On the other hand, a class that is a set may be called a small class. For example, the class of all ordinal numbers and the class of all sets are proper classes.

In general, the paradoxes of naive set theory explains why not all classes can be sets. For example, Russell’s paradox suggests that the class of all sets which do not contain themselves is proper; and Burali-Forti paradox suggests that the class of all ordinal numbers is proper. Hence, the paradoxes tell us that some collections may not be sets. The notion of proper class tells us that we can still talk about those collections even though they are not sets per se. For instance, we can still talk about the collection of all ordinals or prove that some properties hold, despite it is not a set, e.g.

  • The class of ordinal numbers is well-ordered.

Axiomatic Set Theory

Naive set theory may give rise to paradoxes. Axiomatic set theory was originally devised to get rid of such paradoxes. Today, when mathematicians talk about set theory as a field, they usually mean axiomatic set theory. Informal applications of set theory in other fields are sometimes referred to as applications of naive set theory, but are understood to be justified in terms of an axiomatic system.

Let us note that naive set theory may refer to several distinct notions, e.g.

  • Informal presentation of an axiomatic set theory, e.g. as in Naive Set Theory by Paul Halmos.
  • Early or later versions of Georg Cantor’s theory and other informal systems.
  • Decidedly inconsistent theories (whether axiomatic or not), like a theory of Gottlob Frege that yielded Russell’s paradoxes, and theories of Giuseppe Peano and Rechard Dedekind.

References

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