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# Galileo’s paradox

Galileo Galilei (1564—1642) has realised the ostensible contradiction in the situation, that although the set

$1,\,2,\,3,\,4,\,5,\,\ldots$ |

of the positive integers contains all the members of the set

$1,\,4,\,9,\,16,\,\ldots$ |

of the perfect squares and in addition many others, however both sets are equally great in the sense that any member of the former set has as its square a unique counterpart in the latter set and also any member of the latter set has as its square root a unique counterpart in the former set. Galileo explained this by the infinitude of the sets.

In modern mathematical expressions, we say that an infinite set and its proper subset set may have the same cardinality.

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## Mathematics Subject Classification

03E10*no label found*

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