[FOM] Ordinals as hereditarily transitive sets

[Frode Bjørdal]

Sorry, It should have been HT as earlier in the thread.

F​rode Bjø​rdal


On Tue, May 5, 2015 at 2:58 AM, Robert Solovay <solovay at gmail.com> wrote:

> 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).
>>>
>>> Thanks for your consideration.
>>> Gert
>>>
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>>
>>
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