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{{Math-m}}

{{Math-m}}

For example, there is a bijection between the integers and the even integers—it maps every integer x to the even integer 2x.  Notice that this satisfies the definition of bijection.  This means that the integers have the same cardinality as the even integers, even though the former set is "obviously bigger" than the latter.  This leads to the idea, which must be interpreted with a mathematical grain of salt, that "infinity times two is infinity".

For example, there is a bijection between the integers and the even integers—it maps every integer x to the even integer 2x.  Notice that this satisfies the definition of bijection.  This means that the integers have the same cardinality as the even integers, even though the former set is "obviously bigger" than the latter.  This leads to the idea, which must be interpreted with a mathematical grain of salt, that "infinity times two is infinity".

Does this mean that all infinite sets have the same cardinality, that is, can be placed into one-to-one correspondence with each other?  No.  Having the same cardinality as the integers is a very important property, and such sets are called "[[countable]]".  But there are "uncountable" sets that can't be placed into one-to-one correspondence with the integers.  The [[real numbers]] are the best-known example of such.  The proof that the real numbers are not countable is the celebrated "[[Diagonalization|Cantor diagonalization theorem]]".

Does this mean that all infinite sets have the same cardinality, that is, can be placed into one-to-one correspondence with each other?  No.  Having the same cardinality as the integers is a very important property, and such sets are called "[[countable|countable]]".  But there are "uncountable" sets that can't be placed into one-to-one correspondence with the integers.  The [[real numbers]] are the best-known example of such.  The proof that the real numbers are not countable is the celebrated "[[Diagonalization|Cantor diagonalization theorem]]".

[[Category:Mathematics]] [[Category:Set Theory]]

[[Category:Mathematics]] [[Category:Set Theory]]