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that there exists an injective function from X to Y. The Cantor-Bernstein-Schroeder theorem states that if | X | <= | Y | and | Y | <= | X | then | X | = | Y |. The axiom of choice is equivalent to the statement that given two sets X and Y, either | X | <= | Y | or | Y | <= | X |. |
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that there exists an injective function from X to Y. The [Cantor-Bernstein-Schroeder theorem]? states that if | X | <= | Y | and | Y | <= | X | then | X | = | Y |. The axiom of choice is equivalent to the statement that given two sets X and Y, either | X | <= | Y | or | Y | <= | X |. |
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A set X is infinite?, or equivalently, its cardinal is infinite?, if there exists a proper subset Y of X with | X | = | Y |. It can then be proved that the finite cardinals are just the natural numbers, i.e., that a set X is finite if and only if | X | = | n | = n for some natural number n. It can also be proved that the cardinal ω of the set of natural numbers is the smallest infinite cardinal, i.e., that any infinite set admits a subset of cardinality ω. This first infinite cardinal is also often denoted by aleph-0 (where aleph is the first letter in the Hebrew alphabet). The next larger cardinal is denoted by aleph-1 and so on. For every ordinal a there is a cardinal number aleph-a. |
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A set X is infinite?, or equivalently, its cardinal is infinite?, if there exists a proper subset Y of X with | X | = | Y |. A cardinal which is not infinite is called finite; it can then be proved that the finite cardinals are just the natural numbers, i.e., that a set X is finite if and only if | X | = | n | = n for some natural number n. It can also be proved that the cardinal aleph-0 (where aleph is the first letter in the Hebrew alphabet) of the set of natural numbers is the smallest infinite cardinal, i.e., that any infinite set admits a subset of cardinality aleph-0. The next larger cardinal is denoted by aleph-1 and so on. For every ordinal a there is a cardinal number aleph-a, and this list exhausts all cardinal numbers. |
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If X and Y are disjoint, the cardinal of the union of X and Y is called | X | + | Y |. We also define the product of cardinals by | X | x | Y | = | X x Y | (the product on the right hand side is the cartesian product). Also | X || Y | = | XY | where XY is defined as the set of all functions from Y to X. It can be shown that for finite cardinals these operations coincide with the usual operations for natural numbers. For infinite sets (assuming the axiom of choice) we have | X | + | Y | = | X | x | Y | = max{| X |, | Y |}. On the other hand, 2| X | is the cardinality of the power set of the set X and Cantors Diagonal argument shows that 2| X | > | X | for any set X. This proves that there exists no largest cardinal. In fact, the class of cardinals is a proper class. |
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If X and Y are disjoint, the cardinal of the union of X and Y is called | X | + | Y |. We also define the product of cardinals by | X | x | Y | = | X x Y | (the product on the right hand side is the cartesian product). Also | X || Y | = | XY | where XY is defined as the set of all functions from Y to X. It can be shown that for finite cardinals these operations coincide with the usual operations for natural numbers. For infinite sets (assuming the axiom of choice) we have | X | + | Y | = | X | x | Y | = max{| X |, | Y |}. On the other hand, 2| X | is the cardinality of the power set of the set X and Cantors Diagonal argument shows that 2| X | > | X | for any set X. This proves that there exists no largest cardinal. In fact, the class of cardinals is a proper class. |
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The continuum hypothesis states that there are no cardinals strictly between ω and 2ω (the cardinal of the set of real numbers, or the continuum, whence the name). |
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The continuum hypothesis states that there are no cardinals strictly between aleph-0 and 2aleph-0. The latter cardinal number is also often denoted by c; it is the cardinality the set of real numbers, or the continuum, whence the name. |
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