cardinality
Sign in to savethumb|318x318px|A one-to-one correspondence between a set of apples and a set of oranges shows they have the same cardinality. In mathematics, cardinality is an inherent property of sets, roughly meaning the number of individual objects they contain, which may be infinite. The concept is understood through one-to-one correspondences between sets. That is, if their objects can be paired such that each object has a pair, and no object is paired more than once.
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Cardinality is a mathematical property that describes how many individual objects are in a set, whether that number is finite or infinite. It's determined by checking if the objects in two sets can be paired up perfectly—if every object in one set matches with exactly one object in another set with nothing left over, then the sets have the same cardinality.
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Article
36 sectionsContents
- Introduction
- Definition
- Note on notation and terminology
- Etymology and related terms
- Comparing sets
- Equinumerosity
- Equivalence
- Inequality
- Countability
- Countable sets
- Hilbert's hotel
- Uncountable sets
- Cardinal numbers
- Finite sets
- Aleph numbers
- Cardinal arithmetic
- Set of all cardinal numbers
- Cardinality of the continuum
- Skolem's paradox
- Alternative and additional axioms
- Without the axiom of choice
- Proper classes
- Large cardinals
- Constructability
- Determinacy
- History
- Ancient history
- Pre-Cantorian set theory
- Early set theory
- Georg Cantor
- Other contributors
- Axiomatic set theory
- See also
- References
- Citations
- Bibliography
thumb|318x318px|A one-to-one correspondence between a set of apples and a set of oranges shows they have the same cardinality. In mathematics, cardinality is an inherent property of sets, roughly meaning the number of individual objects they contain, which may be infinite. The concept is understood through one-to-one correspondences between sets. That is, if their objects can be paired such that each object has a pair, and no object is paired more than once.
The basic concepts of cardinality go back as early as the 6th century BCE, and there are several close encounters with it throughout history, however, the results were generally dismissed as paradoxical. It is considered to have been first introduced formally to mathematics by Georg Cantor at the turn of the 20th century. Cantor's theory of cardinality was then formalized, popularized, and explored by many influential mathematicians of the time, and has since become a fundamental concept of mathematics.