Where do the natural numbers come from? (Q1202499)

This paper offers a model-theoretic development of the ideas presented in the author's paper: ``Logicism and the ontological commitment of arithmetic'' [J. Philos. 81, No. 3, 123-149 (1984)]. In Section 1, Section 5 and Section 10 I will discuss the philosophical motivation for this project. For more, see the author [loc. cit.; ``Ontological commitment, thick and thin'', in: G. Boolos (ed.), Meaning and method: Essays in honour of Hilary Putnam (Cambridge, 1990)]. More information about some of the higher-order logics that will be introduced below may be found in the author's papers: ``Cardinality logics. I, II'' [Ann. Pure Appl. Logic 39, No. 3, 199-238 (1988; Zbl 0653.03023); J. Symb. Logic 53, No. 3, 765-784 (1988; Zbl 0668.03003)].