## A concrete example of Godel’s Incompleteness theorem [closed]

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,[1] 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?

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 No. In any event, this is not a question of interest to mathematical researchers, as per our faq. The question might be well-received on math.stackexchange – Gerry Myerson Sep 9 2011 at 6:30 Ok, I will try posting there. Thanks. – Nikhil Bellarykar Sep 9 2011 at 6:38 Still, you can have a look at : en.wikipedia.org/wiki/Goodstein%27s_theorem – Adrien Sep 9 2011 at 7:42 Have a look at en.wikipedia.org/wiki/Hilbert's_tenth_problem. In particular, the consequence of the Matiyasevich/MRDP Theorem should be of interest to mathematicians outside of mathematical logic and is easy to state. (you'll have to cut and paste the full link as I can't get it to work properly) – Adam Harris Sep 9 2011 at 8:56 @ Adrian and Adam: Thanks for the links, those helped. – Nikhil Bellarykar Sep 9 2011 at 10:19