Ok, here’s an attempt. There are formal languages. They have many interpretations. You can add axioms to reduce the number of interpretations (as in, if you only count those that make all axioms true). If you add enough, there’s only one interpretation left. This is always possible, but it may require an uncomputable set of axioms. In fact, there’s a result which we could call the difficulty-of-restricting-interpretations-with-axioms-theorem that says that some languages always require uncomputable sets of axioms.
Put like this, it sounds like a pretty boring result, at least to me. But everything about the above description is accurate! The “Incompleteness” theorem is just a result about how many axioms you need to restrict the set of interpretations. Maybe there is a reason why this is important, but the post doesn’t explain this, and I maintain that the sentence “there will always be unprovable truths” (which was the sentence I complained about) is heavily misleading. It’s like a language trick; it sounds impressive because we usually think of truth as being absolute, but here it’s relative. If we did use truth as being absolute (i.e., only call sentences true that are true under every interpretation), the claim is false; in fact, then every true statement is provable—that’s Gödel’s Completeness Theorem.
Ok, here’s an attempt. There are formal languages. They have many interpretations. You can add axioms to reduce the number of interpretations (as in, if you only count those that make all axioms true). If you add enough, there’s only one interpretation left. This is always possible, but it may require an uncomputable set of axioms. In fact, there’s a result which we could call the difficulty-of-restricting-interpretations-with-axioms-theorem that says that some languages always require uncomputable sets of axioms.
Put like this, it sounds like a pretty boring result, at least to me. But everything about the above description is accurate! The “Incompleteness” theorem is just a result about how many axioms you need to restrict the set of interpretations. Maybe there is a reason why this is important, but the post doesn’t explain this, and I maintain that the sentence “there will always be unprovable truths” (which was the sentence I complained about) is heavily misleading. It’s like a language trick; it sounds impressive because we usually think of truth as being absolute, but here it’s relative. If we did use truth as being absolute (i.e., only call sentences true that are true under every interpretation), the claim is false; in fact, then every true statement is provable—that’s Gödel’s Completeness Theorem.