# [FOM] Ordinals as hereditarily transitive sets

Robert Solovay solovay at gmail.com
Tue May 5 01:58:08 EDT 2015

```What does FT mean?

-- Bob Solovay
On May 4, 2015 10:51 PM, "Frode Bjørdal" <frode.bjordal at ifikk.uio.no> wrote:

> 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).
>>
>> Gert
>>
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>
>
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