Re:The Axiom of Choice Is Wrong (2007)
by cygx on Sunday February 05, 2017 @01:43PM
(#13573781)
No incompleteness proves that there will be statements that are true or false that cannot be proven to be true or false.
Careful there: Gödels first incompleteness theorem proves that in a consistent effectively axiomatized formal system, there will be statements that are true or false that cannot be proven to be true or false.
So let's assume there were such a thing as objective mathematical reality and that it could be formalized as The System. As far as I'm aware, all the incompleteness theorem implies is that The System's set of theorems is not recursively enumerable (ie we will never be able to 'write them down').
All we can do is craft necessarily incomplete subsystems applicable to areas of our interest. It's a bit sad that humans will never know the Whole Truth, but such is life...
Careful there: Gödels first incompleteness theorem proves that in a consistent effectively axiomatized formal system, there will be statements that are true or false that cannot be proven to be true or false.
So let's assume there were such a thing as objective mathematical reality and that it could be formalized as The System. As far as I'm aware, all the incompleteness theorem implies is that The System's set of theorems is not recursively enumerable (ie we will never be able to 'write them down').
All we can do is craft necessarily incomplete subsystems applicable to areas of our interest. It's a bit sad that humans will never know the Whole Truth, but such is life...