 The heuristic role of paradoxes is witnessed by Gödel himself, when he intuitively and explicitly relates his construction of formally undecidable sentences to epistemological paradoxes (“the analogy with the Richard antinomy leaps to the eye”, Gödel 1931, van Heijenoort 1967, p. 599). However, self-referential constructions attained an adequate degree of mathematical rigor and became genuine mathematical tools only when non trivial number-theoretic techniques were put to work (see the entry  recursive functions),  for instance in the analysis of syntactical substitution and in providing arithmetical models of formal provability (the crucial role of substitution for producing contradictions was already noticed by Russell, although he did not publish this; see Pelham and Urquhart 1994). Conceptually, it is clear from Gödelian constructions that self-reference by itself is innocuous if it is understood in the indirect sense: one can have formulas \(\phi(x)\), expressing properties of their own “name” \(\ulcorner \phi \urcorner\), but no dangerous circularity arises.