> There's no particular reason to expect any particular formal system to be complete, sans some demonstration that is. Well, Godel's First Incompleteness Thm s

> There's no particular reason to expect any particular formal system to be complete, sans some demonstration that is. Well, Godel's First Incompleteness Thm says that any consistent, effectively axiomatized theory that is strong enough to represent basic arithmetic must be incomplete. So there cannot exist a demonstration of completeness for any such system. ZFC is such a system/theory. (And yes, of course some formal theories are complete and can be demonstrated to be complete, like Presburger arithmetic). > Gödelian incompleteness is the specific kind established by Gödel's proof, where theory T can't prove Con(T) without being inconsistent.... That's the Second Incompleteness Thm. But CH is an example of a (an important; or believed to be important) mathematical statement that cannot be proven / refuted from within ZFC. So it is an example of a statement to which the First Incompleteness Thm applies. You're right that Incompletness #1 does not prove CH is undecidable; (assuming ZFC is consistent) it proves that undecidable sentences must exist. CH is one of those sentences. >...is a similar phenomenon as that the group axioms neither prove nor disprove commutativity I haven't done a course in abstract algebra (though I have studied Incompleteness), but the brief research I just did suggests that there is a meaningful difference in ZFC and the ordinary group axioms. The latter are not sufficiently axiomatized to represent arithmetic, so Godel's thms don't apply. ZFC is sufficiently axiomatized to represent arithmetic (and much more). The ordinary group axioms are so weak that they can describe many different structures; they also cannot represent arithmetic. So the ordinary group axioms are incomplete, but they are not Godel Incomplete.