reference request - Why can't we define more elementary functions?

$\newcommand{\lax}{\operatorname{lax}}$

Liouville's theorem is well known and it asserts that:

The antiderivatives of certain elementary functions cannot themselves be expressed as elementary functions.

The problem I got from this is what is an elementary function? Who defines them? How do we define them?

Someone can, for example, say that there is a function which is called $\lax(\cdot)$ which is defined as:

$$
\lax\left(x\right)=\int_{0}^{x}\exp(-t^2)\mathrm{d}t.
$$

Then, we can say that $\lax(\cdot)$ is a new elementary function much like $\exp(\cdot)$ and $\log(\cdot)$, $\cdots$.

I just do not get elementary functions and what the reasons are to define certain functions as elementary.

Maybe I should read some papers or books before posting this question. Should I? I just would like to get some help from you.

Answer

Elementary functions are finite sums, differences, products, quotients, compositions, and $n$th roots of constants, polynomials, exponentials, logarithms, trig functions, and all of their inverse functions.

The reason they are defined this way is because someone, somewhere thought they were useful. And other people believed him. Why, for example, don't we redefine the integers to include $1/2$? Is this any different than your question about $\mathrm{lax}$ (or rather $\operatorname{erf}(x)$)?

Convention is just that, and nothing more.