When many proofs by contradiction end with "we have built an object with such, such and such properties, which does not exist", it seems relevant to give this object a name, even though (in fact because) it does not exist. The most striking example in my field of research is the following.
Definition : A random variable $X$ is said to be uniform in $\mathbb{Z}$ if it is $\mathbb{Z}$-valued and has the same distribution as $X+1$.
Theorem : No random variable is uniform in $\mathbb{Z}$.
What are the non-existing objects you have come across?
Best Answer
The elliptic curve attached to a nontrivial solution of $x^n+y^n=z^n,\quad n>2$.