That is, the mapping:
- $\forall n \in \N: f: n \mapsto n^2$
is a bijection.
This is a veridical paradox.
$\N$ is an infinite set.
Source of Name
This entry was named for Galileo Galilei.
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.