A model of second-order arithmetic satisfying AC but not DC

TL;DR

This paper constructs a second-order arithmetic model satisfying the Axiom of Choice but not Dependent Choice for certain assertions, revealing limitations of reflection principles in set theory.

Contribution

It demonstrates the existence of a β-model of second-order arithmetic with AC but failing DC for a specific class, confirming a conjecture of Stephen Simpson.

Findings

Existence of a β-model satisfying AC but not DC for a Π^1_2 assertion

Reflection Principle can fail in models of ZFC^-

Rediscovery of a known result by the authors

Abstract

We show that there is a $\beta$-model of second-order arithmetic in which the choice scheme holds, but the dependent choice scheme fails for a $\Pi^1_2$-assertion, confirming a conjecture of Stephen Simpson. We obtain as a corollary that the Reflection Principle, stating that every formula reflects to a transitive set, can fail in models of ${\rm ZFC}^-$. This work is a rediscovery by the first two authors of a result obtained by the third author.