an unstated assumption in Godels Incompleteness Theorem
Exceuse me I h ave come up with a possible way around Godels theorem.
A crucial fact in the theorem is that the theory T (any extension of PA) can encode recursively “x proves y”.
but we well know that there are many fast growing functions that can’t be proved total in PA.. thus...
if we define a theory of mathematics which has a very complicated (algorithm complexity)defintiion of “x proves y” (in ZFC meta-theory for example), so fast growing that it can’t be define in T.
then T may be a theory containing arithmetic for which godels theorem does not apply.. may even a consistent theory T exists than can prove its own consistency!
an unstated assumption in Godels Incompleteness Theorem
Exceuse me I h ave come up with a possible way around Godels theorem.
A crucial fact in the theorem is that the theory T (any extension of PA) can encode recursively “x proves y”.
but we well know that there are many fast growing functions that can’t be proved total in PA.. thus...
if we define a theory of mathematics which has a very complicated (algorithm complexity)defintiion of “x proves y” (in ZFC meta-theory for example), so fast growing that it can’t be define in T.
then T may be a theory containing arithmetic for which godels theorem does not apply.. may even a consistent theory T exists than can prove its own consistency!