The first incompleteness theorem says that for any consistent formal system T (with a recursively enumerable set of axioms) capable expressing of elementary arithmetic, T can express a statement which it can neither prove nor disprove.

The second incompleteness theorem says that T can't prove the statement "T is consistent". (I've still glossed over a number of technical details here; pick up a book on model theory if you want all the messy internals.)

First order logic is notably not capable of expressing elementary arithmetic. And observers aren't involved in any way.