Re:The Axiom of Choice Is Wrong (2007)
by lmm on Sunday February 05, 2017 @01:45PM
(#13573800)
> If they can neither be proved true or false within the system, that means their truth or falsity can be added as axioms to the system and a contradiction will never be reached.
Assuming the original system was consistent (which you can never prove), yes.
But that doesn't help you, because the new system with these additional axioms will still, provably, contain statements that can't be proven true or false within that system (or else be inconsistent). (Turing did try some programme where you keep adding new axioms and take the transitive closure over all the axioms you'd add, but AIUI that doesn't work out either).
Assuming the original system was consistent (which you can never prove), yes.
But that doesn't help you, because the new system with these additional axioms will still, provably, contain statements that can't be proven true or false within that system (or else be inconsistent). (Turing did try some programme where you keep adding new axioms and take the transitive closure over all the axioms you'd add, but AIUI that doesn't work out either).