Galileo's Paradox/Historical Note

Historical Note on Galileo's Paradox
initially raised this paradox in his (Discourses and Mathematical Demonstrations Relating to Two New Sciences) of $1638$, during his discussions of continuity.

The question was raised again by in his posthumous ($1851$)  (The Paradoxes of the Infinite), where he indicated that he was perfectly comfortable with it, and gave many other examples of how an infinite set is equivalent to a proper subset of it.

and later used this property of an infinite set to actually define the concept.

From this philosophical standpoint, then went on to create his theory of infinite cardinals.