FOM: foundations using functions

Randall Holmes holmes at
Wed Jan 21 13:21:48 EST 1998

Thanks to Machover for calling everyone's attention to von Neumann's
axioms of 1925 (see p. 393 of van Heijenoort's _From Frege to
G\"odel_).  They express more or less the kind of foundation for a
ZFC-like theory that I had in mind, except that proper class functions
are also provided.  In any event, both sets and classes can be
interpreted in this function-based system as functions taking on two
values, one of which is a special value taken on "almost everywhere"
by functions which are capable of being arguments (i.e., which would
be sets rather than classes in a formulation based on membership
instead of application).

Another nice feature of von Neumann's system is the fact that it
avoids appealing to the notion "formula of first-order logic".  Sets
witnessing instances of separation or replacement are constructed
using a finite set of operations.

And God posted an angel with a flaming sword at | Sincerely, M. Randall Holmes
the gates of Cantor's paradise, that the       | Boise State U. (disavows all) 
slow-witted and the deliberately obtuse might | holmes at
not glimpse the wonders therein. |

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