[FOM] Ordinal definable numbers
ali.enayat at gmail.com
Fri Jul 23 22:10:12 EDT 2010
About two months ago (June 10), Tim Chow posed the following question
to FOM subscribers, a question that was originally posed by Garabed
Gulbenkian on MathOverflow.net, see
>Is the following statement consistent with ZFC?
>"There exists a denumerably infinite and ordinal definable set of real
>numbers, not all of whose elements are ordinal definable."
The point of this posting is twofold:
(1) The statement "There exists a definable set of real numbers that
is Dedekind finite, but none of whose members is ordinal definable" is
consistent with ZF (not ZFC!); and
(2) To suggest a model of ZFC that will very likely yield a positive
answer to Garabedian's original question.
(1) follows from combining two results - one by Solovay, the other by
Kanovei . Both concern the symmetric submodel N of the generic
extension M[G] corresponding to adding countably many side-by-side
"Jensen generic reals" to a model M of ZFC.
For an expository account of Jensen-generic reals see Section 28 of
Jech's Set Theory (Millenium edition). Here let me just say that
Jensen reals are Delta^1_3 definable and minimal. The notion of
forcing that produces them is obtained by "thinning" Sacks forcing
using the diamond principle.
Solovay showed that the collection of Jensen reals in N is definable.
Kanovei, on the other hand, observed that the same collection is
Solovay's proof appears in the following paper of mine as Theorem 3.3.
On the Leibniz-Mycielski Axiom in Set Theory , Fundamenta
Mathematicae, vol. 181 (2004), pp.215-231.
Now regarding (2). The model is very likely to yield a positive answer
to the original question is the generic extension M[G] corresponding
to adding countably many side-by-side "Jensen generic reals" to a
model M of ZFC.
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