[FOM] Ordinals as hereditarily transitive sets

Gert Smolka smolka at ps.uni-saarland.de
Sun May 3 15:43:43 EDT 2015

I have a historical question.  I would like to know where the following characterizations of von Neumann ordinals appeared first:

An ordinal is a transitive set all whose elements are transitive.

Let us refer to this characterization as HT for hereditarily transitive.  HT assumes a set theory where the membership relation is well-founded (regularity).

I looked at some historical papers and can say the following.

1) Von Neumann's first definition of ordinals in 1923 is rather roundabout: Given a WO, one obtains the ordinal for this WO by transfinite recursion.

2) Von Neumann' second definition of ordinals in 1928 in a paper on definition by transfinite induction is more explicit: An ordinal is a well-ordered set x such that for all y in x, y is the set of all z in x such that z < y.  This definition is used by Halmos 1960.

3) Robinson 1937 gives an explicit definition of ordinals as plain sets:  An ordinal is a transitive set x such that for all y,z in x either y in x or y=x or x in y.

4) Kunen and many others define ordinals as transitive sets that are well-ordered by the membership relation.

5) HT appears as Exercise I.7.26 in Kunen's book "Set Theory" (2011).

6) HT appears in Forster's book Reasoning about theoretical entities (2003).

7) Alain Badiou in his book "Number and Numbers" (2008) defines ordinals using HT (Chapter 8).

8) To the best of my knowledge, HT appears neither in the mentioned papers of von Neumann (1913, 1928) nor in Robinson's paper (1937).

Thanks for your consideration.

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