Galileo's Paradox of Infinity

Visualize the one-to-one correspondence between natural numbers and perfect squares.

Last updated: June 2026 | By Patchworkr Team

Infinite Mapping

Range

Choose a range between 5 and 20 to visualize the pairing between natural numbers and their perfect squares.

Result

Natural numbers

1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Perfect squares

1, 4, 9, 16, 25, 36, 49, 64, 81, 100

1 -> 1, 2 -> 4, 3 -> 9, 4 -> 16, 5 -> 25, 6 -> 36, 7 -> 49, 8 -> 64, 9 -> 81, 10 -> 100

What the paradox shows

A subset can have the same cardinality as the full set when both are infinite.

Cardinality measures the size of a set. For infinite sets, a bijection proves equal size.

n to n^2

Example: 1, 2, 3, 4 map to 1, 4, 9, 16.

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