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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 ...
, the concept of
cardinality The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...
is significantly developable without recourse to actually defining
cardinal numbers In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the case ...
as objects in the theory itself (this is in fact a viewpoint taken by Frege; Frege cardinals are basically
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
es on the entire
universe The universe is all of space and time and their contents. It comprises all of existence, any fundamental interaction, physical process and physical constant, and therefore all forms of matter and energy, and the structures they form, from s ...
of sets, by equinumerosity). The concepts are developed by defining equinumerosity in terms of functions and the concepts of one-to-one and
onto In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...
(injectivity and surjectivity); this gives us a quasi-ordering relation :A \leq_c B\quad \iff\quad (\exists f)(f : A \to B\ \mathrm) on the whole universe by size. It is not a true partial ordering because antisymmetry need not hold: if both A \leq_c B and B \leq_c A, it is true by the Cantor–Bernstein–Schroeder theorem that A =_c B i.e. ''A'' and ''B'' are equinumerous, but they do not have to be literally equal (see
isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
). That at least one of A \leq_c B and B \leq_c A holds turns out to be equivalent to the
axiom of choice In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...
. Nevertheless, most of the ''interesting'' results on cardinality and its
arithmetic Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms. ...
can be expressed merely with =c. The goal of a cardinal assignment is to assign to every set ''A'' a specific, unique ''set'' that is only dependent on the cardinality of ''A''. This is in accordance with Cantor's original vision of 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 \leq_c, 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 A model is an informative representation of an object, person, or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin , . Models can be divided int ...
of set theory. In modern set theory, we usually use the
Von Neumann cardinal assignment The von Neumann cardinal assignment is a cardinal assignment that uses ordinal numbers. For a well-orderable set ''U'', we define its cardinal number to be the smallest ordinal number equinumerous to ''U'', using the von Neumann definition of a ...
, which uses the theory of
ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the leas ...
s and the full power of the axioms of
choice A choice is the range of different things from which a being can choose. The arrival at a choice may incorporate Motivation, motivators and Choice modelling, models. Freedom of choice is generally cherished, whereas a severely limited or arti ...
and replacement. Cardinal assignments do need the full axiom of choice, if we want a decent cardinal arithmetic and an assignment for ''all'' sets.


Cardinal assignment without the axiom of choice

Formally, assuming the axiom of choice, the cardinality of a set ''X'' is the least ordinal ''α'' such that there is a
bijection In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
between ''X'' and ''α''. This definition is known as the
von Neumann cardinal assignment The von Neumann cardinal assignment is a cardinal assignment that uses ordinal numbers. For a well-orderable set ''U'', we define its cardinal number to be the smallest ordinal number equinumerous to ''U'', using the von Neumann definition of a ...
. 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 The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by the mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1 ...
'') is as the set of all sets that are equinumerous with ''X'': this does not work in ZFC or other related systems of
axiomatic 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 ...
because this collection is too large to be a set, but it does work in
type theory In mathematics and theoretical computer science, a type theory is the formal presentation of a specific type system. Type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a foundation of ...
and in New Foundations and related systems. However, if we restrict from this
class Class, Classes, or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used d ...
to those equinumerous with ''X'' that have the least rank, then it will work (this is a trick due to
Dana Scott Dana Stewart Scott (born October 11, 1932) is an American logician who 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, C ...
: it works because the collection of objects with any given rank is a set; see Scott's trick).


References

*Moschovakis, Yiannis N. ''Notes on Set Theory''. New York: Springer-Verlag, 1994. {{Mathematical logic Cardinal numbers Set theory