$\mathbb{C}$ is nice and safe, in that it's algebraically closed, so we can solve all polynomials in $\mathbb{C}$, but there are still equations that cannot be solved in this set.
I'm wondering: Can we construct a set of numbers in which all uni-variable equations (so $0=1$ doesn't count; nor does $= \begin{cases} x+y=1 \\ x+y=2 \end{cases}$) have at least one solution.
This includes, but is not restricted to, the following examples:
- $x=x+1$ (obviously, if such a set exists, then we cannot cancel things additively)
- $e^x=0$
- $\frac{1}{x}=0$
- $\sin(x)=2$
- $|x|<0$
- $\sin^2(x)+\cos^2(x)=10$
I understand that, if this 'magic' set exists, addition, multiplication and exponentiation (amongst other operations) won't be commutative, associative, or even alternative, that we have to allow zero divisors and that it won't obey the usual rules of algebra.
Also, what unusual properties would this set have, were it to exist?
Let $X$ be a set with more than $1$ element and take $a,b \in X$ with $a\not = b$. Consider the function $f(x) \equiv a$, then the equation $$f(x) = b$$ doesn't have a solution.