# Galileo’s paradox

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

$$1,\mathrm{\hspace{0.17em}2},\mathrm{\hspace{0.17em}3},\mathrm{\hspace{0.17em}4},\mathrm{\hspace{0.17em}5},\mathrm{\dots}$$ |

of the positive integers all the members of the set

$$1,\mathrm{\hspace{0.17em}4},\mathrm{\hspace{0.17em}9},\mathrm{\hspace{0.17em}16},\mathrm{\dots}$$ |

of the perfect squares^{} and in 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 , we say that an infinite set^{} and its proper subset^{} set may have the same cardinality.

Title | Galileo’s paradox^{} |
---|---|

Canonical name | GalileosParadox |

Date of creation | 2013-03-22 19:15:45 |

Last modified on | 2013-03-22 19:15:45 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 6 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 03E10 |

Related topic | Paradox |