Oracle bites Theory

Abstract

In the context of Elementary Arithmetic (EA) we know that already an extremely weak arithmetical theory like R proves every true Σ1-sentence. Thus, it would seem that adding the true Σ1-sentences to the axiom set of a given theory adds nothing. However, Elementary Arithmetic cannot prove this ‘obvious fact’. We show that under the assumption of the negation of Σ1-collection, the weak theory PA− plus the true Σ1-sentences is inconsistent. It follows that, in EA plus the negation of Σ1-collection, any consistent extension U of PA− is not closed under finite conjunctions: there is a conjunction of theorems of U such that U plus that conjunction is inconsistent. A corollary of our main insight is that Σ1-collection is, over Elementary Arithmetic, equivalent to the restricted consistency of PA− plus the true Σ1-sentences. In Appendix C, we prove slightly modified results for the weaker theory R. A consequence of these results is that, over EA, Σ1-collection is equivalent to the consistency of the theory axiomatized by the theorems of R. In Appendix A, we provide a lay person’s summary of the results of the paper.