&gt;&gt; Then Gödel shows up and says, &quot;On which shelf shall we put the index?&quot;<p>&gt; You keep it outside the library. The index tells you where every single book is, other than the index.<p>This approach would have saved Russell and Whitehead&#x27;s &quot;Principia Mathematica&quot; project, would have placed mathematics on a firm foundation, but it suffers from the same defect of self-reference -- it places the index beyond validation. Russell realized this, and recognized that Gödel&#x27;s results invalidated his project.<p>&gt; The fact that Coq cannot be proven in Coq does not decrease our confidence that it is correct;<p>Yes, true, but for the reason that we have no such confidence (and can have no such confidence). We cannot make that assumption, because the Turing Halting problem and its theoretical basis prevents it.<p>Source: <a href="http:&#x2F;&#x2F;www.sscc.edu&#x2F;home&#x2F;jdavidso&#x2F;math&#x2F;goedel.html" rel="nofollow">http:&#x2F;&#x2F;www.sscc.edu&#x2F;home&#x2F;jdavidso&#x2F;math&#x2F;goedel.html</a><p>Quote: &quot;Gödel hammered the final nail home by being able to demonstrate that <i>any consistent system of axioms would contain theorems which could not be proven.</i> Even if new axioms were created to handle these situations this would, of necessity, create new theorems which could not be proven. <i>There&#x27;s just no way to beat the system.</i> Thus he not only demolished the basis for Russell and Whitehead&#x27;s work--that all mathematical theorems can be proven with a consistent set of axioms--but he also showed that no such system could be created.&quot; (emphasis added)