In the

philosophy of mathematics The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in peop ...

, specifically the philosophical foundations of

set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...

, limitation of size is a concept developed by

Philip Jourdain Philip Edward Bertrand Jourdain (16 October 1879 – 1 October 1919) was a British logician and follower of Bertrand Russell. Background He was born in Ashbourne in Derbyshire* one of a large family belonging to Emily Clay and his father Franc ...

and/or

Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ; – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of ...

to avoid

Cantor's paradox In set theory, Cantor's paradox states that there is no set of all cardinalities. This is derived from the theorem that there is no greatest cardinal number. In informal terms, the paradox is that the collection of all possible "infinite sizes" is ...

. It identifies certain "inconsistent multiplicities", in Cantor's terminology, that cannot be sets because they are "too large". In modern terminology these are called proper classes.

Use

The

axiom of limitation of size In set theory, the axiom of limitation of size was proposed by John von Neumann in his 1925 axiom system for sets and classes.; English translation: . It formalizes the limitation of size principle, which avoids the paradoxes encountered in earli ...

is an axiom in some versions of

von Neumann–Bernays–Gödel set theory In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel–choice set theory (ZFC). NBG introduces the notion of class, which is a colle ...

or Morse–Kelley set theory. This axiom says that any class that is not "too large" is a set, and a set cannot be "too large". "Too large" is defined as being large enough that the class of all sets can be mapped one-to-one into it.

References

* Philosophy of mathematics History of mathematics Basic concepts in infinite set theory {{settheory-stub