The resolution of Galileo's paradox came centuries later through the groundbreaking work of Georg Cantor in the late 19th century. Cantor developed a rigorous mathematical framework for dealing with infinite sets, introducing the concept of cardinality that could handle infinite quantities systematically.
Formal Definition of Cardinality
In Cantor's framework, two sets have the same cardinality if and only if there exists a bijection (one-to-one correspondence) between them. This definition works perfectly for finite sets but becomes the key to understanding infinite sets. The cardinality of the natural numbers is denoted by ℵ₀ (aleph-null), the smallest infinite cardinal number.
For Galileo's paradox, we define the bijection f: ℕ → S where S is the set of perfect squares, by f(n) = n². This function is: (1) Injective: if f(a) = f(b), then a² = b², which implies a = b; (2) Surjective: for every perfect square s, there exists n = √s such that f(n) = s; (3) Therefore bijective, proving |ℕ| = |S|.
Cantor's Insight and Modern Understanding
Cantor's revolutionary insight was that the property of being a proper subset is not incompatible with having the same cardinality in infinite sets. In fact, this becomes a defining characteristic: a set is infinite if and only if it can be put in one-to-one correspondence with a proper subset of itself. This is known as the Dedekind-infinite property.
The natural numbers, integers, rational numbers, and perfect squares all have the same cardinality ℵ₀, making them countably infinite. However, Cantor also proved that the real numbers have a strictly larger cardinality (2^ℵ₀), demonstrating that there are indeed different 'sizes' of infinity.