NB: Abstract added by Editors. In his contribution, Bob Hale aims at providing a theory of ordinals on neo-logicist bases. In this respect, two main goals have to be accomplished. First and foremost, consistency is to be retained, since Burali-Forti’s paradox easily arises from a naïve ordinal abstraction principle accompanied by unrestricted second-order logic. Second, such a reconstruction should not be provided on the basis of set theory but rather in a purely neo-logicist setting. Hale achieves both goals by modifying Shapiro and Wright’s definition of indefinite extensibility in Shapiro and Wright (2006), on the one hand, and relying on Hume’s Principle, on the other.