>[!abstract] >**Galileo's paradox** is a mathematical [[Gedanken|thought experiment]] demonstrating that infinite sets defy the common-sense laws of finite counting. It shows that you can pair two infinite sets together perfectly, even if one set is entirely contained within the other. In his *magnus opus* titled *Discorsi e dimostrazioni matematiche intorno a due nuove scienze* (*Discourses and Mathematical Demonstrations Relating to Two New Sciences*), Galileo examined the following two sets of natural numbers: $S_1$ is the set of all natural numbers, and $S_2$ is the set of all perfect squares (i.e., the squares of all natural numbers). Galileo's first intuition is that there are fewer square numbers than natural numbers, so $S_1$ must be larger than $S_2$: ``` S1 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ...} S2 = {1, 4, 9, 16, ...} ``` His second intuition, however, is that every natural number can be paired with its square, without any leftover in either set: ``` S1 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...} ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ S2 = {1, 4, 9, 16, 25, 36, 49, 64, 81, 100, ...} ``` From which he concluded that normal size comparisons (like "larger", "smaller", or "equal") simply cannot be applied to infinite quantities. Cantor developed this idea further to form modern set theory. Cantor resolved the paradox by formally establishing that infinite sets can be measured using cardinality. Crucially, all countably infinite sets (those that can be placed in a one-to-one correspondence) such as $S_1$ and $S_2$, are of the same cardinality. >[!related] >- **North** (upstream): — >- **West** (similar): — >- **East** (different): — >- **South** (downstream): —