In set theory, the concept of cardinality is significantly developable without recourse to actually defining cardinal numbers as objects in the theory itself (this is in fact a viewpoint taken by Frege; Frege cardinals are basically equivalence classes on the entire universe of sets, by equinumerosity). 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 quasi-ordering relation A ≤ c B ⟺ ( ∃ f ) ( f : A -> B i s i n j e c t i v e ) {\displaystyle A\leq _{c}B\quad \iff \quad (\exists f)(f:A\to B\ \mathrm {is\ injective} )} on the whole universe by size. It is not a true partial ordering because antisymmetry need not hold: if both A ≤ c B {\displaystyle A\leq _{c}B} and B ≤ c A {\displaystyle B\leq _{c}A} , it is true by the Cantor–Bernstein–Schroeder theorem that A = c B {\displaystyle A=_{c}B} i.e. A and B are equinumerous, but they do not have to be literally equal (see isomorphism). That at least one of A ≤ c B {\displaystyle A\leq _{c}B} and B ≤ c A {\displaystyle B\leq _{c}A} holds turns out to be equivalent to the axiom of choice. 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 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 ≤ c {\displaystyle \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 of set theory. 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. ## Cardinal assignment without the axiom of choice[edit] Formally, assuming the axiom of choice, the 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 that 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). ## References[edit] * Moschovakis, Yiannis N. Notes on Set Theory. New York: Springer-Verlag, 1994. * v * t * e Mathematical logic General| * Axiom * list * Cardinality * First-order logic * Formal proof * Formal semantics * Foundations of mathematics * Information theory * Logical consequence * Model * Set * Theorem * Theory * Type theory Theorems (list) & Paradoxes| * Gödel's completeness and incompleteness theorems * Tarski's undefinability * Banach–Tarski paradox * Cantor's theorem, paradox and diagonal argument * Compactness * Halting problem * Lindström's * Löwenheim–Skolem * Russell's paradox Logics| | Traditional| * Classical logic * Logical truth * Tautology * Proposition * Inference * Logical equivalence * Consistency * Equiconsistency * Argument * Soundness * Validity * Syllogism * Square of opposition * Venn diagram | Propositional| * Boolean algebra * Boolean functions * Logical connectives * Propositional calculus * Propositional formula * Truth tables * Many-valued logic * 3 * Finite * ∞ Predicate| * First-order * Second-order * Monadic * Higher-order * Free * Quantifiers * Predicate * Monadic predicate calculus Set theory| * Set * Hereditary * Class * (Ur-)Element * Ordered pair * Ordinal number * Subset * Equality * Extensionality * Forcing * Relation * Equivalence * Partition * Set operations: * Intersection * Union * Complement * Cartesian product * Power set * Identities Types of Sets| * Countable * Uncountable * Empty * Inhabited * Singleton * Finite * Infinite * Transitive * Ultrafilter * Recursive * Fuzzy * Universal * Universe * Constructible * Grothendieck * Von Neumann Maps & Cardinality| * Function/Map * Domain * Codomain * Image * In/Sur/Bi-jection * Schröder–Bernstein theorem * Isomorphism * Gödel numbering * Enumeration * Large cardinal * Inaccessible * Aleph number * Operation * Binary Set theories| * Zermelo–Fraenkel * Axiom of choice * Continuum hypothesis * General * Kripke–Platek * Morse–Kelley * Naive * New Foundations * Tarski–Grothendieck * Von Neumann–Bernays–Gödel * Constructive Syntax & Language| * Alphabet * Arity * Automata * Axiom schema * Expression * Ground * Extension * Relation * Formal * Grammar * Language * Proof * System * Theory * Formation rule * Formula * Atomic * Closed * Ground * Open * Free/bound variable * Metalanguage * Logical connective * ¬ * ∨ * ∧ * → * ↔ * = * Predicate * Functional * Variable * Propositional variable * Quantifier * ∃ * ! * ∀ * rank * Sentence * Atomic * Signature * String * Substitution * Symbol * Function * Logical/Constant * Non-logical * Variable * Term Example axiomatic systems (list)| * of arithmetic: * Peano * second-order * elementary function * primitive recursive * Robinson * Skolem * of the real numbers * Tarski's axiomatization * of Boolean algebras * canonical * minimal axioms * of geometry: * Euclidean * Elements * Hilbert's * non-Euclidean * Tarski's * Principia Mathematica Proof theory| * Formal proof * Natural deduction * Logical consequence * Rule of inference * Sequent calculus * Theorem * Systems * Formal * Axiomatic * Deductive * Hilbert * list * Complete theory * Independence (from ZFC) * Proof of impossibility * Ordinal analysis * Reverse mathematics * Self-verifying theories Model theory| * Interpretation * Model * Equivalence * Finite * Saturated * Substructure * Non-standard model * of arithmetic * Diagram * Elementary * Categorical theory * Model complete theory * Satisfiability * Semantics of logic * Strength * Theories of truth * Semantic * Tarski's * Kripke's * T-schema * Transfer principle * Truth predicate * Truth value * Type * Ultraproduct * Validity Computability theory| * Church encoding * Church–Turing thesis * Computably enumerable * Computable function * Computable set * Decision problem * Decidable * Undecidable * P * NP * P versus NP problem * Kolmogorov complexity * Lambda calculus * Primitive recursive function * Recursion * Recursive set * Turing machine * Type theory Related| * Abstract logic * Category theory * Concrete/Abstract Category * Category of sets * History of logic * History of mathematical logic * timeline * Logicism * Mathematical object * Philosophy of mathematics * Supertask Mathematics portal *[v]: View this template *[t]: Discuss this template *[e]: Edit this template