Set of numbers that are solutions for equations such as $ x = x + 1 $

Question

Is there any number set that is used in that kind of equations without solution on the complex numbers?

Answer 1

Extended real numbers. We add two numbers to the real numbers namely $-\infty,\infty$ with $c\infty=\infty$ if $c>0$ and $-\infty$ if $c<0$. $\infty+\infty=\infty$ , $\infty-\infty$ undefined. Check

https://en.m.wikipedia.org/wiki/Extended_real_number_line

Clearly if you work with in $G$ for some set and you want to check the solution of and you each element have an inverse w.r.t $+$ then

$\begin{array}{ccc}x&=&x+1 \\ x-x&=&-x+x+1\\ 0&=&1\end{array}$

hence in our $G$ , we have $1=0$ i.e. $1$ is the addition inverse. But am sure you are not looking for something like this right?