In the

philosophy of mathematics Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship to other areas of philosophy, particularly epistemology and metaphysics. Central questions posed include whether or not mathem ...

, specifically the philosophical foundations of

set theory Set theory is the branch of mathematical logic that studies Set (mathematics), 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 mathema ...

, limitation of size is a concept developed by Philip Jourdain and/or

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

to avoid Cantor's paradox. 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 is an axiom in some versions of von Neumann–Bernays–Gödel set theory 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