English Dictionary

Galileo's paradox

固有名詞

(set theory) A demonstration of a surprising property of infinite sets. Some positive integers are squares while others are not; therefore, all the numbers, including both squares and non-squares, must be more numerous than just the squares; yet for every square there is exactly one positive number that is its square root, and for every number there is exactly one square; hence, there cannot be more of one than of the other.

日本語の意味

無限集合の意外な性質を示す逆説。具体的には、正の整数全体と、その平方数との間に一対一対応が存在するという事実を示し、部分集合であっても元の集合と同じ濃度を持つというパラドックス。 / ガリレオの逆説。集合論における無限集合の性質に注目し、数字の平方数との対応関係から、一見矛盾する性質が明らかになる現象を表す。

In his writings on mathematics, Galileo's paradox highlights how the set of perfect squares and the set of all positive integers can be put into a one-to-one correspondence despite the former seeming smaller.

数学に関する著作で、無限集合が真部分集合と一対一対応できるという驚くべき性質を示す議論は、平方数の集合と全ての正の整数の集合が、前者が小さく見えるにもかかわらず一対一対応できることを強調している。

意味（1）

(set theory) A demonstration of a surprising property of infinite sets. Some positive integers are squares while others are not; therefore, all the numbers, including both squares and non-squares, must be more numerous than just the squares; yet for every square there is exactly one positive number that is its square root, and for every number there is exactly one square; hence, there cannot be more of one than of the other.