To be precise, let $\Omega$ be an inaccessible cardinal in $L$ and let N be the Solovay model defined by the Levy-collapse in this case. Then $\Omega$ is $\aleph_1$ in $N$.

How many different OD (=ordinal-definable) sets of reals are there in $N$?

One answers that there is exactly $\aleph_2$ of them.

Now, is there a concrete, meaningful OD sequence of $\aleph_2$ different OD sets of reals in $N$?

  • $\begingroup$ What sets of reals are in $\rm OD\setminus HOD$? $\endgroup$ – Asaf Karagila Jun 17 '14 at 19:26
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    $\begingroup$ And welcome to the site! $\endgroup$ – Asaf Karagila Jun 17 '14 at 19:30
  • $\begingroup$ Sets in OD are those definable by a set theoretic formula with ordinals as parameters. This contains, as a small part, e.g. sets in "lightface" projective classes $\Sigma^1_n$. HOD means that the set itsefl, all its elements, all elements of elements etc are OD. For instance the set R of all reals is OD but not HOD (generally speaking) $\endgroup$ – Vladimir Kanovei Jun 17 '14 at 21:16
  • $\begingroup$ Right, of course. My bad. Thank you for pointing that out! $\endgroup$ – Asaf Karagila Jun 17 '14 at 22:25

We can start with a canonical sequence of $\Omega^+$ many distinct subsets of $\Omega$ in $L$, which is a sequence of $\aleph_2$ many distinct subsets of $\aleph_1$ in $N$. Then we can use the canonical partial surjection $\mathbb{R} \to \aleph_1$, which takes a real coding a well-ordering of $\omega$ to the order type of this well-ordering, to get a sequence of $\aleph_2$ many distinct subsets of $\mathbb{R}$ in $N$.

I think this example is about as concrete and meaningful as we can hope for, because we will need to use somehow the fact that the ground model satisfies $2^\Omega = \Omega^+$.


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