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People always bring up the Continuum Hypothesis as though it has something to do with Gödel's incompleteness theorems, but it doesn't really. The Continuum Hypo

by Chinjut 19d ago

People always bring up the Continuum Hypothesis as though it has something to do with Gödel's incompleteness theorems, but it doesn't really. The Continuum Hypothesis being neither proven nor disproven by some particular axioms is a similar phenomenon as that the group axioms neither prove nor disprove commutativity, the ordered field axioms neither prove nor disprove the existence of a square root of 2, etc. There's no particular reason to expect any particular formal system to be complete, sans some demonstration that is. Gödelian incompleteness is the specific kind established by Gödel's proof, where theory T can't prove Con(T) without being inconsistent. But the inability of ZFC to consistently decide the Continuum Hypothesis is established in a completely different manner, with the Continuum Hypothesis not consistently decided by ZFC + Con(ZFC) either, or any such thing.

jsrozner 19d ago
> 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.

by Chinjut 19d ago

jsrozner 19d ago

> 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.