# Galileo's Paradox

## 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

This is a veridical paradox.

$\N$ is an infinite set.

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

## Also see

## 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.

## Sources

- 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.33$: Weierstrass ($1815$ – $1897$): Footnote $5$