The 19th-century mathematician who later gave a framework for comparing infinite sets was Georg Cantor. Galileo’s paradox observes that the positive integers and the perfect squares can be paired one-to-one, even though the squares seem to form only a part of the integers.
Galileo concluded in Two New Sciences that ordinary ideas of greater and lesser did not work straightforwardly for infinite collections. In the 19th century, Cantor developed a systematic theory of sets and cardinality. His approach treated two sets as equally large when their members could be matched bijectively, making the integers and squares the same size in that sense.
Cantor also showed that not all infinities are equal: the real numbers are uncountable, while the integers are countable. The common mix-up is to credit Galileo with modern set theory; Galileo supplied an important early example, but Cantor created the later mathematical framework.