Are there thoughtfully simple concepts that we cannot currently prove?

Question

I was driving and just happened to wonder if there existed some concepts that are simple to grasp, yet are not provable via current mathematical techniques. Does anyone know of concepts that fit this criteria?

I imagine the level of simple could vary considerably from person to person, myself being on the very low end of things.

Answer 1

The simplest that I know of, I don't know if you consider this simple to grasp, is the Continuum Hypothesis.

$\textbf{The statement}$: there is no infinity between the cardinality of $\mathbb{Z}$ ($|\mathbb{Z}|=\aleph_0$) and the cardinality of $\mathbb{R}$.

It has been proven that this can neither be proven nor disproven with Zermelo-Fraenkel set theory with the axiom of choice.

Answer 2

we know that $e$ is transcendental, $\pi$ is transcendental. but still we dont know that whether $e + \pi$ and $e - \pi$ is transcendental or not. we know that atleast one of them is transcendental which follows from simple calculation.