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.