What is a Proof?
&gt; Near the turn of the 20th century, some important advancements were made in mathematical logic. Principia Mathematica was an attempt to report the axioms and inference rules needed to prove any mathematical truth. 20 years later, Gödel’s incompleteness theorem proved that such a goal could never be accomplished.<p>This part fascinates me. As I understand it Gödel&#x27;s incompleteness theorem states essentially that &quot;in any formal axiomatic system large enough to contain arithmetic there will be true statements that cannot be proven&quot;. But doesn&#x27;t that apply to the less formal &quot;natural language&quot; proofs of mathematicians as well...? When I read sections like the above (in many different places) I sometimes get the feeling that the authors think that &quot;Gödel proved that formal proofs cannot prove all mathematical truths, so we mathematicians switched to informal proofs instead (to get around Gödel&#x27;s incompleteness theorem and continue on our quest to prove all mathematical truths)&quot;. Can someone confirm that this is <i>not</i> the intended meaning? :P<p>As I see it, let there be sets of mathematical statements A, B and C such that A are all true statements, B are all statements that can be formally proven and C are all statements that have a beautiful proof (from &quot;The Book&quot;); then C must be a (possibly equal) subset of B, which must in turn be a (possibly equal) subset of A. Right? :)