 where \(B \approx C\) is short-hand for one of the many equivalent formulas of pure second order logic expressing the equivalence relation “there is a one to one correlation between the objects falling under \(B\) and those falling under \(C\)”. Informally, it can be read as saying that two concepts \(B\) and \(C\) have the same ‘number’ if and only if there is a one to one correspondence among the objects that fall under \(B\) and those that fall under \(C\). Principles like HP that define a function from an equivalence relation are called abstraction principles. By presupposing the existence of a function that sends concepts into objects, Hume’s Principle exploits the possibility of defining infinitely many concepts that do not stand in the equivalence relation mentioned in its right-hand side to generate infinitely many natural numbers. There are other varieties of neologicism that do not postulate Hume’s Principle or an axiom of infinity at the outset but yield infinitely many natural numbers by means of other logical principles (see for instance Linsky and Zalta 2006). In addition, all these varieties of neologicism generate at least an infinite cardinal numbers and what is of philosophical relevance here is the different resources they use to establish these results.