Galileo’s paradox

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

1, 2, 3, 4, 5,

of the positive integers all the members of the set

1, 4, 9, 16,

of the perfect squaresMathworldPlanetmath 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 setMathworldPlanetmath and its proper subsetMathworldPlanetmathPlanetmath set may have the same cardinality.

Title Galileo’s paradoxMathworldPlanetmath
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