Galileo's Paradox of Infinity Calculator

Galileo's paradox begins with two observations that seem incompatible. Perfect squares are only some of the positive whole numbers: 1, 4, 9, 16, and so on. Most natural numbers are not squares, and finite samples make the squares appear increasingly sparse. Yet every positive natural number n pairs with exactly one square n squared, and every positive square has exactly one positive square root. This one-to-one correspondence suggests that the set of squares and the set of all natural numbers have the same infinite size. The calculator makes the finite side visible. Among the natural numbers from 1 through a limit N, the number of perfect squares is floor(sqrt(N)). Through 100 there are 10 squares, so squares make up 10% of the sample. Through 10,000 there are 100, only 1%. As N grows, the proportion floor(sqrt(N))/N approaches zero. In the language of natural density, perfect squares have density zero among the natural numbers. The infinite side uses pairing rather than proportion. Match 1 with 1, 2 with 4, 3 with 9, and in general n with n squared. No natural number on the input side is omitted, and no perfect square on the output side is repeated or omitted. Modern set theory calls such a rule a bijection. Two sets have the same cardinality when a bijection exists between them, even when one set is a proper subset of the other. Both sets are countably infinite. There is no contradiction once finite proportion and infinite cardinality are recognized as different measurements. Proportion asks how densely one collection occurs inside another up to growing cutoffs. Cardinality asks whether members can be paired one for one. For finite sets these ideas align with familiar expectations: a proper subset always has fewer members. Infinite sets behave differently because they can pair completely with proper subsets of themselves. Galileo discussed this puzzle before a rigorous theory of infinite cardinal numbers existed. Later, Georg Cantor formalized one-to-one correspondence as the way to compare infinite sets. The paradox remains a useful introduction to countable infinity, limits, density, and bijections. Experiment with progressively larger limits to watch the finite percentage shrink, while remembering that the mapping n to n squared never runs out on either side. The tool does not calculate infinity; it illustrates how finite trends and infinite matching answer two distinct mathematical questions.