## Theorem

The natural numbers $\N$ are in one-to-one correspondence with their squares:

That is, the mapping:

$\forall n \in \N: f: n \mapsto n^2$

is a bijection.

Hence the set of square numbers is equinumerous to the set of natural numbers.

That is, a set is equinumerous to one of its proper subsets.

## Resolution

$\N$ is an infinite set.

A defining property of an infinite set is that it possesses proper subsets with which it is equinumerous.

## Source of Name

This entry was named for Galileo Galilei.

## Historical Note

Galileo Galilei initially raised this paradox in his Discorsi e Dimostrazioni Matematiche Intorno a Due Nuove Scienze (Discourses and Mathematical Demonstrations Relating to Two New Sciences) of $1638$, during his discussions of continuity.

The question was raised again by Bernhard Bolzano in his posthumous ($1851$) Paradoxien des Unendlichen (The Paradoxes of the Infinite), where he indicated that he was perfectly comfortable with it, and gave many other examples of how an infinite set is equivalent to a proper subset of it.

Richard Dedekind and Georg Cantor later used this property of an infinite set to actually define the concept.

From this philosophical standpoint, Cantor then went on to create his theory of infinite cardinals.