If X is a set of formulas and A is a formula then we say that A is  ⊢-deducible  from  X, or “X  ⊢  A”, if there exists a sequence A0, A1, …, An of formulas such that An = A and for all i≤n, Ai is an instance of an axiom schema, or a formula of X, or comes from earlier formulas of the sequence by a rule of inference. Further,  ⊢  A iff ∅  ⊢  A; in this case we say that A is  ⊢-deducible.  X is said to be  ⊢-consistent  iff not X  ⊢  0. It is easy to establish that (I) can be replaced by the following axiom schema (induction axiom schema):