Nobody is practically concerned with the “incompleteness” aspect of Gödel’s theorems. The unprovable statements are so pathological/contrived that it doesn’t appear to suggest any practical statement might be unprovable. Consistency is obviously more important. Sufficiently weak systems may also not be limited by the incompleteness theorems, i.e. they can be proved both complete and consistent.
Oh, what if the Riemann hypothesis is such a statement then? Or any other mathematical statement. We may not have any use for them now, but as with all things math, they are sometimes useful somewhere unexpected.
Math is also used to make a statement/model our universe. And we are still trying to find the theory to unify quantum mechanics and gravity. What if our math is simply inconsistent hence the theory of everything is not possible within the current mathematical framework?
Sure when you are solving the problems it is useless to ponder about it, but it serves as a reminder to also search for other ideas and not outright dismiss any strange new concept for a mathematical system. Or more generally, any logical system that follows a set of axioms. Just look at the history of mathematics itself. How many years before people start to accept that yes imaginary numbers are a thing.
Dunno what you’re trying to say. Yes, if ZFC is inconsistent it would be an issue, but in the unlikely event this is discovered, it would be overwhelmingly probable that a similar set of axioms could be used in a way which is transparent to the vast majority of mathematics. Incompleteness is more likely and less of an issue.
Just explaining that the limitations of Gödel’s theorems are mostly formal in nature. If they are applicable, the more likely case of incompleteness (as opposed to inconsistency) is not really a problem.
It’s very counter intuitive. As the other commenter suggested I was referring to Gödel and his incompleteness theorem.
Actually if the system you made up doesn’t work it would be possible to prove that it does inside that system as you can prove anything inside a system that doesn’t work.
That is why my comment is not entirely accurate it should actually be: Until you prove that if the system works you can’t prove that the system works.
Until you prove that you can’t prove that the system you made up works.
Nobody is practically concerned with the “incompleteness” aspect of Gödel’s theorems. The unprovable statements are so pathological/contrived that it doesn’t appear to suggest any practical statement might be unprovable. Consistency is obviously more important. Sufficiently weak systems may also not be limited by the incompleteness theorems, i.e. they can be proved both complete and consistent.
Oh, what if the Riemann hypothesis is such a statement then? Or any other mathematical statement. We may not have any use for them now, but as with all things math, they are sometimes useful somewhere unexpected.
It’s extremely unlikely given the pathological nature of all known unprovable statements. And those are useless, even to mathematicians.
Math is also used to make a statement/model our universe. And we are still trying to find the theory to unify quantum mechanics and gravity. What if our math is simply inconsistent hence the theory of everything is not possible within the current mathematical framework?
Sure when you are solving the problems it is useless to ponder about it, but it serves as a reminder to also search for other ideas and not outright dismiss any strange new concept for a mathematical system. Or more generally, any logical system that follows a set of axioms. Just look at the history of mathematics itself. How many years before people start to accept that yes imaginary numbers are a thing.
Dunno what you’re trying to say. Yes, if ZFC is inconsistent it would be an issue, but in the unlikely event this is discovered, it would be overwhelmingly probable that a similar set of axioms could be used in a way which is transparent to the vast majority of mathematics. Incompleteness is more likely and less of an issue.
I think the statement “this system is consistent” is a practical statement that is unprovable in a sufficiently powerful consistent system.
Can you help me understand the tone of your text? To me it sounds kinda hostile as if what you said is some kind of gotcha.
Just explaining that the limitations of Gödel’s theorems are mostly formal in nature. If they are applicable, the more likely case of incompleteness (as opposed to inconsistency) is not really a problem.
Then it doesn’t work
No, see Gödels Incompleteness theorem
It’s very counter intuitive. As the other commenter suggested I was referring to Gödel and his incompleteness theorem.
Actually if the system you made up doesn’t work it would be possible to prove that it does inside that system as you can prove anything inside a system that doesn’t work.
That is why my comment is not entirely accurate it should actually be: Until you prove that if the system works you can’t prove that the system works.
Can you spot the difference in the logic here?