What kind of algebraic equations do trandescendal numbers not solve?

Question

I know transcendental numbers cannot solve polynomials or rational functions (since they can always be written as a polynomial), but are they the solutions to equations containing a variable raised to a non-integer?

Answer 1

Whatever you accept as generalization of polynomials, I assume the candidate expressions can be expressed with finitely many symbols taken from a finite alphabet (e.g. it is possible to write them down in an intelligible manner on paper or using $\LaTeX$).. Moreover I assume you only accept functions that have at most countably many zeroes in $\mathbb R$ (so this allows us to define $\pi$ as the smallest positive zero of $\sin x$, for example). Then you still can catch at most countably many numbers, hence certainly not all transcendentals.