In set theory, the concept of cardinality is significantly developable without recourse to actually defining cardinal numbers as objects in theory itself (this is in fact a viewpoint taken by Frege; Frege cardinals are basically equivalence classes on the entire universe of sets which are equinumerous). The concepts are developed by defining equinumerosity in terms of functions and the concepts of one-to-one and onto (injectivity and surjectivity); this gives us a pseudo-ordering relation Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ... Alternative meaning: number of pitch classes in a set. ... Friedrich Ludwig Gottlob Frege (8 November 1848, Wismar – 26 July 1925, Bad Kleinen) was a German mathematician who evolved into a logician and philosopher. ... Two sets A and B are said to be equinumerous if they have the same cardinality, i. ... Two sets A and B are said to be equinumerous if they have the same cardinality, i. ... In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ...

on the whole universe by size. It is not a true ordering because the trichotomy law need not hold: if both and , it is true by the Cantor–Bernstein–Schroeder theorem that A = _cB i.e. A and B are equinumerous, but they do not have to be literally equal; that at least one case holds turns out to be equivalent to the Axiom of choice. Generally, a trichotomy is a splitting into three disjoint parts. ... In set theory, the Cantor–Bernstein–Schroeder theorem, named after Georg Cantor, Felix Bernstein, and Ernst Schröder, states that, if there exist injective functions f : A → B and g : B → A between the sets A and B, then there exists a bijective function h : A → B. In terms of... In mathematics, the axiom of choice, or AC, is an axiom of set theory. ...

Nevertheless, most of the interesting results on cardinality and its arithmetic can be expressed merely with =_c.

The goal of a cardinal assignment is to assign to every set A a specific, unique set which is only dependent on the cardinality of A. This is in accordance with Cantor's original vision of a cardinals: to take a set and abstract its elements into canonical "units" and collect these units into another set, such that the only thing special about this set is its size. These would be totally ordered by the relation and =_c would be true equality. As Y. N. Moschovakis says, however, this is mostly an exercise in mathematical elegance, and you don't gain much unless you are "allergic to subscripts." However, there are various valuable applications of "real" cardinal numbers in various models of set theory. Georg Cantor Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845, St. ... In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the models which underlie mathematical systems. ...

In modern set theory, we usually use the Von Neumann cardinal assignment which uses the theory of ordinal numbers and the full power of the Axioms of choice and replacement. Cardinal assignments do need the full Axiom of choice, if we want a decent cardinal arithmetic and an assignment for all sets. More on this (and much more good set theory in general!) can be found in Moschovakis' excellent introduction to set theory. The von Neumann cardinal assignment is a cardinal assignment which uses ordinal numbers. ... In mathematics, the axiom of choice, or AC, is an axiom of set theory. ... In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of replacement is a schema of axioms in Zermelo-Fraenkel set theory. ...

Cardinal assignment without the axiom of choice

Formally, assuming the axiom of choice, cardinality of a set X is the least ordinal α such that there is a bijection between X and α. This definition is known as the von Neumann cardinal assignment. If the axiom of choice is not assumed we need to do something different. The oldest definition of the cardinality of a set X (implicit in Cantor and explicit in Frege and Principia Mathematica) is as the set of all sets which are equinumerous with X: this does not work in ZFC or other related systems of axiomatic set theory because this collection is too large to be a set, but it does work in type theory and in New Foundations and related systems. However, if we restrict from this class to those equinumerous with X that have the least rank, then it will work (this is a trick due to Dana Scott: it works because the collection of objects with any given rank is a set). The von Neumann cardinal assignment is a cardinal assignment which uses ordinal numbers. ... The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910-1913. ... The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ... Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. ... At the broadest level, type theory is the branch of mathematics and logic that concerns itself with classifying entities into collections called types. ... In mathematical logic, New Foundations (NF) is a candidate set theory proposed by Willard van Orman Quine, obtained from a streamlined version of the theory of types of Bertrand Russell. ... In mathematical set theory, the rank of a set is recursively defined as the smallest ordinal number greater than the rank of any member of the set. ... Dana Stewart Scott (born 1932) is the emeritus Hillman University Professor of Computer Science, Philosophy, and Mathematical Logic at Carnegie Mellon University; he is now retired and lives in Berkeley, California. ...

Reference

• Moschovakis, Yiannis N. Notes on Set Theory. New York: Springer-Verlag, 1994.