Erratum to: Localizing the axioms

In clause (i) of Lemma 2.1 of [1] it is claimed that in BST we can show the existence of ω as the least inductive set. BST contains the axiom of Infinity saying that “there is an inductive set”. However one cannot see how to prove the existence of a least inductive set without either ∈-induction or at least Π1-Separation, both of which are not included in BST. The simplest way to correct this flaw is to replace the above Infinity axiom with the stronger version: “There is a smallest inductive set, which we call ω”. Then clause (i) of Lemma 2.1 is modified as follows: “The axioms of Peano arithmetic hold in ω endowed with the usual operations. Thus PA ⊆ BST.” Also the proof of clause (i) is modified as follows: “The minimality of ω as inductive set amounts to the fact that ω satisfies complete induction. The operations ′, +, · on it are defined as usual and the axioms of PA are shown in BST to be true with respect to ω.” The above stronger version of Infinity axiom is a Σ2 sentence (while the old version was Σ1). This slightly affects the truth of Remark 2.5, where it is claimed that all axioms of BST are Π2 sentences.