# [FOM] Ordinals as hereditarily transitive sets

Frode Bjørdal frode.bjordal at ifikk.uio.no
Sun May 3 20:33:25 EDT 2015

```As there is a lack of references to literature in the previous millennium I
point out that the definition FT of ordinals is presupposed in Lemma 23.4
in Andre Cantini's monograph *Logical Frameworks for Truth and Abstraction
- An Axiomatic Study*, Elsevier 1996. Certainly this definition HT was
first given at some earlier time.

..........................................
Professor Dr. Frode Bjørdal
Universitetet i Oslo Universidade Federal do Rio Grande do Norte
quicumque vult hinc potest accedere ad paginam virtualem meam
<http://www.hf.uio.no/ifikk/personer/vit/fbjordal/index.html>

On Sun, May 3, 2015 at 4:43 PM, Gert Smolka <smolka at ps.uni-saarland.de>
wrote:

> 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.
> Gert
>
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