FOM: Zermelo's successor function references

[Jeff Hirst]

>Dear FOM'ers,
>
>I underdstand that E. Zermelo first suggested representing Peano's successor
>function by the set theoretic operation of forming singletons (?). Can
>anyone on the list verify this; if so, is a citation available?
>
>Many Thanks,
>
>Peter A.
>Toronto

Hi Peter-

One article in which Zermelo made such a suggestion is:

E. Zermelo, Untersuchungen uber die Grundlagen der Mengenlehre I,
Math. Ann. 65 (1908), 261--281.

This article appears in translated form as:

Investigations in the foundations of set theory I
in the (excellent) source book:
J. van Heijenoort, From Frege to Godel, Harvard University Press,
Cambridge, 1967.

The particular information you're interested in follows Zermelo's
presentation of Axiom VII (Axiom of Infinity) and appears on pages
204--205 of the Hiejenoort volume.  Zermelo's formulation of the
Axiom of Infinity asserts the existence of a set Z which contains
the null set and also contains the set {a} whenever it contains a.
In the following paragraph he proves the existence of a set Z_0
which "is the common component of all possible sets constituted
like Z. ...  The set Z_0 contains the elements 0, {0}, {{0}}, and
so forth, and it may be called the number sequence, because its
elements can take the place of the numerals."

Hope this helps...

-Jeff
-- 
Jeff Hirst   jlh at math.appstate.edu
Professor of Mathematics
Appalachian State University, Boone, NC  28608
vox:828-262-2861    fax:828-265-8617