Exercise 11, § 3

Find the error in the following argument : Let $\mathbf{N}$ denote the set of natural numbers and let A denote the set of all integers $n > 2$ for which there exist three strictly positive integers $x$, $y$, $z$ such that $x^n + y^n = z^n$. Then the set A is not empty (in other words, “Fermat’s last theorem” is false). For let $\mathrm{B} = \{\mathrm{A}\}$ and $\mathrm{C} = \{\mathbf{N}\}$; B and C are sets consisting of a single element, hence there is a bijection $f$ of B onto C. We have $f(\mathrm{A}) = \mathbf{N}$; if A were empty we should have $\mathbf{N} = f(\emptyset) = \emptyset$, which is absurd. \*