# [FOM] ordinals

G Aldo Antonelli antonelli at ucdavis.edu
Wed Sep 10 13:39:46 EDT 2008

Aki Kanamori wrote:

> (c) In the second of that series, appearing in the JSL in 1941,
> section 5 sets out the theory of ordinals, and Bernays relies
> on the more perspicuous definition of Raphael Robinson:
> An ordinal is a transitive set x which is connected: for
> different members a,b of x, either a \in b or b \in a.
> This first appeared in a JSL paper of Robinson's of 1937, which presented
> the von Neumann class-set theory in simplified form. In Bernays'
> section 5, item 2) presents the equivalence of `hereditarily
> transitive' to Robinson's definition.

Robinson's definition has the obvious advantage of not depending on
foundation, a fact that has more than a passing interest for those of us
who dabble in non-standard, ill-founded, or otherwise just plain weird
set theories.

-- Aldo

*****************************************
G. Aldo Antonelli
Professor of Philosophy
University of California, Davis
Coordinating Editor, Review of Symbolic Logic
http://philosophy.ucdavis.edu/antonelli
antonelli at ucdavis.edu