Random Number Sequences- Do they really exist ?

Well, the big question is - is it possible to create a truly random sequence of numbers ?

I was thinking on these lines -

1. Generation can be either done synthetically (human created algorithm) or by observing a certain parameter of a naturally occuring event. I believe that all human attempts are unlikely to be truly random creations as there shall always be a mathematical logic behind it, which shall always give predictable sequence.
2. The queer part is that, if we believe that eventually science (which always has a mathematical formulation) can explain all natural phenomenon, then we cannot get a sequence of random numbers from nature. (if (1) above is true!)

Does above mean truly random number sequences do not exist. Because if they do then, boy, does it prove that science cannot explain nature in its entirity ?

What do you think ?