|
Session 32. Set Theory
|
On rigidity and weak covering for \(\hbox{HOD}\) |
Sy-David Friedman, Kurt Gödel Research Center, Austria
|
|
|
I'll discuss two results concerning Gödel's universe \(\hbox{HOD}\) of hereditarily
ordinal definable sets, the first supporting the view that \(\hbox{HOD}\) is a
close approximation to the universe \(V\) of all sets and the second
supporting the opposite view. In [2] it was shown that
\(V\) is generic over \((\hbox{HOD},S)\) where \(S\) is the \(V\)-definable
stability predicate. Our first result extends this to class
theory using the \(V\)-definable enriched stability predicate
\(S^*\). A corollary is that \((\hbox{HOD},S^*)\) is rigid with respect to
"V-constructible" embeddings. Our second result, joint with
Cummings and Golshani [1], provides a model in which
\(\alpha^+\) of \(\hbox{HOD}\) is less than \(\alpha^+\) for all infinite
cardinals \(\alpha\).
|
|
|
References- J. Cummings, M. Golshani, S. Friedman,
Collapsing the cardinals of \(\hbox{HOD}\), sumbitted.
- S. Friedman, The stable core , Bulletin of
Symbolic Logic vol.18, no.2, 2012, 261-267.
|
|
| Print version |
|
