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That's right. You could list all those propositions and search for proofs of them. In fact this is very similar to Hilbert's very program to which Godel's First

That's right. You could list all those propositions and search for proofs of them. In fact this is very similar to Hilbert's very program to which Godel's First Incompleteness Thm was a response (https://en.wikipedia.org/wiki/Hilbert%27s_program https://en.wikipedia.org/wiki/Hilbert%27s_program). Godel showed that there are true statements that cannot be proven, and also that among the unprovable statements from within the system is the consistency of the system itself. As I understand, mathematicians are still trying to figure out how much this matters. One of the best examples of its mattering is may be the Continuum Hypothesis: CH is consistent with ZFC, and ~CH (not CH) is also consistent with ZFC. In other words, you have enough flexibility in constructing your ZFC world such that in some ZFC-consistent worlds CH is true, and in others CH is false. Litt's statement is not wrong; it's just that what he wrote sounds so much like Hilbert's program, that I'm surprised we didn't get some even minor comment on what kinds of truths we could reach if we embarked on such an effort.

Chinjut 17d 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 16d 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.

Chinjut 17d 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 16d 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.