# [FOM] Another historical query

Akihiro Kanamori aki at math.bu.edu
Mon Sep 8 09:09:09 EDT 2008

Dear Thomas,

>The von Neumann ordinals are precisely the (wellfounded) hereditarily
>transitive sets.   Does anyone on this list know who first proved it?  Or
>who first published it?

My reponse:

(a) That the ordinals are exactly the hereditarily transitive sets
(in the presence of Foundation) is first stated in
a letter from Paul Bernays to Kurt Godel of 3 May 1931. See
the Godel Collected Works, vol IV.

(b) The letter sets out Bernays's version of the von Neumann
class-set theory. As the Godel-Bernays correspondence makes clear,
this letter is the source of Godel's later adoption of Bernays' system
in his 1940 monograph on L --- not Bernays' later series of papers in the JSL.

(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.

(d) At the end of that item, Bernays wrote that Godel
defined the ordinals as the hereditarily transitive sets in
1937 lectures in Vienna. However, this formulation does not
appear in the Godel 1940 monograph. So, returning full circle
to the May 1931 letter, the formulation should be accredited
to Bernays, with the first published proof in his 1941 paper.

--Best wishes, Aki Kanamori
`