All things that (Neo-Fregean) Numbers could be Vincenzo Ciccarelli Campinas State University Department of Philosophy (IFCH) ciccarelli.vin@gmail.com Keywords: Neo-Fregeanism, Hume’s principle, Benacerraf identification problem Abstract. In this paper I describe a difficulty for the Neo-Fregean account of the concept of natural number. In particular, I prove that, according to the theory developed by Hale and Wright, all objects arrangeable in an infinite progression should be natural numbers. This conclusion, undesirable from the Neo-Fregean standpoint, is drawn by using two fundamental criteria: Hume’s principle conceived as a criterion of application for the concept of natural number and Hale’s criterion of identity for truth-conditions. Since the former principle is formulated in terms of an identity of truth-conditions the natural proposal would be to embed the latter in it. I use this resulting formulation of the criterion application for the concept of number to carry out my proof. In the concluding section I try to suggest along which lines the Neo-Fregean may argue to overcome the difficulty. 1 1 Introduction One of the central claims of the Neo-Fregean view as advocated by Hale and Wright is that the concept of natural number is genuinely sortal. According to the account given in Wright (1983), a concept F is genuinely sortal if and only if F admits both a defined criterion of identity and a defined criterion of application. The former criterion fixes the truth-conditions of the statement ’a = b’ where ’a’ and ’b’ are terms referring to instances of F , whereas the latter fixes the condition for a singular term ’a’ to refer to an instance of F . Moreover, Wright claims that among all genuine sortals the concept of natural number is of a special kind: for its criterion of identity – traditionally known as Hume’s principle – plays also the role of a criterion of application. To understand how this could be, consider the following meta-linguistic version of Hume’s principle: 1 Research for this paper has been supported by a grant from FAPESP (Process n. 2014/27057-5) 1