Paradoxical set

In set theory, a paradoxical set is a set that has a paradoxical decomposition. A paradoxical decomposition of a set is a partitioning of the set into subsets with an appropriate group of functions that operate on the elements of the set, such that each subset can be mapped back into the entire set. Since a paradoxical set as defined requires a suitable group ${\displaystyle G}$, it is said to be ${\displaystyle G}$-paradoxical, or paradoxical with respect to ${\displaystyle G}$.

Paradoxical sets exist as a consequence of the Axiom of Infinity. Admitting infinite classes as sets is sufficient to allow paradoxical sets.

Example

An example of a paradoxical set is the natural numbers. They are paradoxical with respect to the group of functions ${\displaystyle \{{\frac {x}{2}},{\frac {x-1}{2}}\}}$ .

Split the natural numbers into the odds and the evens. If you apply the function ${\displaystyle f(x)={\frac {x-1}{2}}}$  to the odds and ${\displaystyle f(x)={\frac {x}{2}}}$  to the evens; each of these will give back the entire set of the natural numbers.