Godel's incompleteness theorem says "Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory."
I liked the theorem, but had a hard time finding an example. Here, I am proposing an example for Godel's theorem. Please tell me if it is correct or point out the flaws.
Consider an axiomatic system where all the regular axioms regarding real valued functions hold. In particular, this system is concerned with integrals. One additional constraint is that existence of any integral is 'provable' if the indefinite integral can be expressed in terms of elementary functions. Now, given the fact that the integral for error function converges i.e. existence is true, but can't be expressed in terms of elementary functions i.e. not provable, can I say that this is an example wherein a statement is known to be true, but not provable?