top | item 18035895 (no title) atmanthedog | 7 years ago Unfortunately, math doesn't really permit this type of truth: if your axioms are strong enough to prove general statements about arithmetic, there is no effective procedure to determine whether an arbitrary proof follows from those axioms. discuss order hn newest smadge|7 years ago Did you mean to write “there is no effective procedure to determine whether an arbitrary formula follows from those axioms?”A proof is exactly how we demonstrate that a formula follows from the axioms. rocqua|7 years ago Still, given a set of axioms, statements will fall into one of three categories. 1) Provably True, 2) Provably False, 3) UndecideableClaims that a statement is in category 1 are fully verifiable (by providing the proof). The same goes with claims that a statement is in category 2.
smadge|7 years ago Did you mean to write “there is no effective procedure to determine whether an arbitrary formula follows from those axioms?”A proof is exactly how we demonstrate that a formula follows from the axioms.
rocqua|7 years ago Still, given a set of axioms, statements will fall into one of three categories. 1) Provably True, 2) Provably False, 3) UndecideableClaims that a statement is in category 1 are fully verifiable (by providing the proof). The same goes with claims that a statement is in category 2.
smadge|7 years ago
A proof is exactly how we demonstrate that a formula follows from the axioms.
rocqua|7 years ago
Claims that a statement is in category 1 are fully verifiable (by providing the proof). The same goes with claims that a statement is in category 2.