Actually, it IS possible to prove some things can’t be done. It’s been proven that:
You can’t square the circle (i.e. using only a compass and a straight-edge, you can’t make a square with the same area as a circle.)
You can’t find any three integers X, Y and Z where X^3+Y^3=Z^3 holds true.
You can’t write a program that can calculate the last digit of pi.
You can’t determine the exact speed and exact location of a sub-atomic particle at the same time.
You can’t fit a full-grown elephant into a 8oz teacup without compressing it into degenerate matter such as found on a neutron star.
Need I go on?