The key to e.g. Goedel's results is the Fixed Point Lemma: if ψ is a formula with v free then there is a sentence φ such that, provably, φ <=> ψ(<φ>) [where <..> is the numerical code of .. ].

The proof, if you are interested, is not difficult. Let sub be the function that describes, via codes, substituting the (numeral n_ of the number) n for a free variable v of a formula χ : sub(<χv>, n_) = <χn_>. Then the PROOF is: consider ψ(sub(v,v)), call it θv, let m be <θv> and let φ be θm_. Then, provably, φ <=> ψ(sub(m_,m_)) <=> ψ(sub(<θv>,m_)) <=> ψ(<θm_>) <=> ψ(<φ>). Ta-da!

Goedel used this to get the "formula that says I am not provable", φ <=> not Prov(<φ>).