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 all the members of the set

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

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 GalileosParadox 2013-03-22 19:15:45 2013-03-22 19:15:45 pahio (2872) pahio (2872) 6 pahio (2872) Definition msc 03E10 Paradox