Ordinals by Abstraction

The neo-Fregean programme in the philosophy of mathematics seeks to provide foundations for fundamental mathematical theories in abstraction principles. Ian Rumfitt (2018) proposes to introduce ordinal numbers by means of an abstraction principle, (ORD), which says, roughly, that ‘the ordinal number attaching to one well-ordered series is identical with that attaching to another if, and only if, the two series are isomorphic’. Rumfitt’s proposal poses a sharp and serious challenge to those seeking to advance the neo-Fregean programme, for Rumfitt proposes to save (ORD) from threatening paradox by avoiding dependence on an impredicative comprehension principle. However, such a principle is usually taken to be required by the neo-Fregean account of the cardinal numbers. Thus if neo-Fregean foundations for elementary arithmetic are to be saved, we must explain how we can avoid paradox for (ORD) in another way. In this chapter, the prospects for doing so are explored.