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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]]".

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

[[Category:Set theory]]