FOM: Re: Zermelo's successor function

[A.P. Hazen]

Zermelo was willing, in the 1908 paper included in Van Heijenoort, to call
the sereis {},{{}},{{{}}},... "the number series," but he did not
immediately go on (in the manner of modern elementary set theory texts) to
give a relative interpretation of an axiomatic arithmetic in set theory on
this basis: that actually came some time later.  In 1909 he published
treatments of the theory of FINITE SETS.
A good survey of the history here is Charles Parsons's "Developing
Arithmetic in Set Theory without Infinity: some historical remarks," in the
journal "History and Philosophy of Logic," vol. 8 (1987), pp. 201-213.
David Lewis's monograph "Parts of Classes" gives a reformulation of set
theory with the singleton operator rather than the membership relation as a
primitive.  He shows that the axiomatization (in a higher order logic) of
set theory with this primitive is a remarkably close parallel to the
Dedekind/Peano axiomatization of arithmetic based on successor, and so
makes the interpretation of successor as singleton seem less arbitrary than
it usually does.
--
Allen Hazen
Philosophy Department
University of Melbourne