[FOM] which sets numbers can be

M. Randall Holmes holmes at diamond.boisestate.edu
Wed Sep 24 12:06:22 EDT 2003

Responding to a remark by Steinhart:

>It's astonishing that anybody would think that any set theoretic =
>model of the Dedekind-Peano axioms is as good as any other.  The =
>advantages of the von Neumann ordinals are elaborated in almost =
>every introductory text.  Here's just a few quick and dirty =
>examples of advantages of the von Neumann's: (1) the VNs identify =
>less-than with membership; (2) the VNs identify =
>less-than-or-equal-to with subset-inclusion; (3) each VN ordinal is =
>internally well-ordered; (4) the relation that internally =
>well-orders each VN ordinal is the relation that well-orders the =
>whole series of VN ordinals; (4) the VNs are uniformly extendible =
>to the transfinite; (5) the n-th VN ordinal has n members.   =20

I agree with the first sentence!  There are good implementations
and bad implementations.

The Frege natural numbers, in a set theory which admits them, also
have nice features.  The definitions of the basic arithmetic
operations are more straightforward with the von Neumann numbers (no appeal
to recursion is needed, nor is induction needed to prove most theorems
of basic arithmetic).  Frege natural numbers generalize to
Russell-Whitehead cardinal numbers (and the definitions of arithmetic
operations on Frege naturals are inherited by the general cardinals as

This is not to deny that the von Neumann naturals are also a nice

Mathematics is a religion!	|   --Sincerely, M. Randall Holmes
We are Reform Pythagoreans      |   Math. Dept., Boise State Univ.
(we eat beans).	No official BSU |   holmes at math.boisestate.edu
endorsement of above opinions!  |   http://math.boisestate.edu/~holmes

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