This article is confused and its confusion is apparent right from the first paragraph: "In the course of exploring their universe, mathematicians have occasiona

This article is confused and its confusion is apparent right from the first paragraph: "In the course of exploring their universe, mathematicians have occasionally stumbled across holes: statements that can be neither proved nor refuted with the nine axioms, collectively called “ZFC,” that serve as the fundamental laws of mathematics. Most mathematicians simply ignore the holes, which lie in abstract realms with few practical or scientific ramifications. But for the stewards of math’s logical underpinnings, their presence raises concerns about the foundations of the entire enterprise." Actually, it's confused from the first sentence: the job of a mathematician isn't to explore "the universe," which is ultimately an empirical question. Anyhow, what Gödel proved is that such "holes" — a term a mathematician would never use — are a necessary consequence of any sufficiently powerful theory. In a nutshell, Gödel's incompleteness theorem says that if we have an axiomatic system powerful enough to describe the natural numbers and arithmetic — what is called Peano arithmetic — but manageable enough for us to enumerate the axioms then there will be theorems expressible in that axiomatic system which are neither provable nor disprovable in that axiomatic system. Note that they would well be provable or disprovable in some larger, more powerful axiomatic system. Or, like Euclid's parallel postulate, we could have a consistent mathematics whether we assume the theorem or its negation. Even if we extend ZF(C) with axioms which allow us to prove or disprove CH, there will still be other theorems which are unprovable. Because CH is independent of ZF(C), the following statement is true: "If ZF(C) is is consistent then both ZF(C)+CH and ZF(C)+not-CH are consistent." That is, if we believe ZF(C) is consistent, we can add either CH its negation to our set of axioms and remain consistent. This is what it means for CH to be independent of ZF(C). It's also worth noting that the Axiom of Choice — the (C) in ZF(C) — is independent of ZF. This means one can consistently do math in both ZF(C) and ZF(not-C), assuming ZF itself is consistent. In the history of mathematics, Cantor believed CH to be true and Gödel believe it to be false. Nowadays, most mathematicians don't worry about this stuff. Instead, mathematicians prove statements like "In ZF(C), statement X is true if and only if CH is true." Whether one assumes CH or not-CH has to be decided on other grounds, e.g., aesthetics, utility, and so on. The independence of CH from ZF(C) makes arguing for its truth or falsity a fool's errand. I think the error is in believing that every statement in mathematics must be a statement about our universe, i.e., either CH is true or it isn't and that's that. It's not a statement about our universe, however — it's a statement about some axiomatic system we've developed. If that seems weird to you, consider that you play with utterly un-physical constructs all the time, like numbers and perfect circles.