What Gödel Discovered
I thought I replied to this post, but I guess not and my reply ended up being a top-level reply, which is just as well. I will just mention my main two quibbles to this otherwise excellent post and others like it so that other people who embark on introductory posts to Godel&#x27;s results don&#x27;t fall into the same trap.<p>1. Please don&#x27;t bring the notion of truth into an introductory explanation (such as in the section &quot;Power of Numbers&quot; or in the short intro to &quot;Hilbert&#x27;s Program&quot;). It makes it very confusing, especially in light of Godel&#x27;s other landmark result, i.e Godel&#x27;s Completeness Theorem which simultaneously applies to many of the same logical systems and can be very very vaguely described as &quot;all true things are provable.&quot; Just stick to incompleteness and consistency at a syntactic level instead of appealing to truth, that is respectively the ability to always add new axioms and the lack of syntactic contradictions.<p>2. The lack of a system S&#x27;s ability to prove its own consistency is not interesting by itself since we would not trust a proof from a potentially inconsistent system. Hence it is not, in a sense, terribly interesting that we require an infinite tower of increasingly stronger systems to prove the consistency of S and themselves. Rather the interesting direction goes the other way. S cannot prove the consistency of stronger systems, which means we cannot have a &quot;minimal&quot; system we use to explore the consistency of all other systems (again subject to some nuance).