4 ms·
Nonsensical if it's not provable with respect to what theory exactly? Elementary function Arithemetic? Primitive Recursive Arithemetic? Peano Arithemtic? Marti
by rssoconnor 5y ago
Nonsensical if it's not provable with respect to what theory exactly? Elementary function Arithemetic? Primitive Recursive Arithemetic? Peano Arithemtic? Martin-Löf type theory? ZF set theory? ZFC+"there exist an infinite number of Woodin cardinals"? "The set of true statements of number theory"?
Each of these logical theories are each able to prove an increasing number of arithmetic propositions. What is or is not provable is relative the deduction system or selection of axioms.
For example, that big expression that I linked to is designed so that isn't provable in Peano Arithmetic, but it will be provable Martin-Löf type theory, ZF set theory, etc.
- scotty79 5y agoIf it's not provable and not false in F it's nonsensical in F. Might become sensible with more axioms. I'd like to ask you a question... Can you take the expression you linked to and add its negation as the additional axiom to Peano arithmetic? Or would it lead to some contradictions?