&gt; A mathematical proof applies to all cases, not just one. Gödel&#x27;s result addressed Russell&#x27;s project, but it applies universally, as do all valid mathematical results.<p>What? I have no idea what you mean by &quot;applies universally&quot; and &quot;all valid mathematical results.&quot;<p>A straightforward reading of that sentence implies that Pythagoras&#x27; Theorem is not a &quot;valid mathematical result&quot; because it applies only to right triangles. That can&#x27;t be what you&#x27;re saying, is it?<p>I am acknowledging that Gödel&#x27;s second theorem states that any mathematical system that attempts to prove itself is doomed to failure. I am also stating that Russell&#x27;s project involved a mathematical system that attempts to prove itself. Therefore, Gödel&#x27;s second theorem applies to Russell&#x27;s project, but it does not necessarily say anything else about <i>other projects.</i><p>&gt; Many mathematicians regard Gödel&#x27;s result as the most important mathematical finding of the 20th century.<p>So what? This is <i>argumentum ad populum</i>. How important the mathematical finding is is irrelevant to whether it matters to the discussion at hand.<p>&gt; This is false. If the Halting problem is insoluble, it is also insoluble to a human<p>OK, so do you believe that modern mathematics is invalid because Russell&#x27;s project failed?