A well known result (stated and credited to Todorcevic in "Semiselective Coideals", by Farah, Mathematika, 1997, but with antecedents going back to Mathias) says that, under the appropriate large cardinal hypothesis (enough to get all sets of reals in $L(\mathbb{R})$ to be universally Baire, say), a selective ultrafilter is $L(\mathbb{R})$-generic for $([\omega]^\omega,\subseteq^*)$.

It is also well-known that selective ultrafilters need not exist; Kunen showed that they are destroyed by iterating random forcing over a model of CH. More generally, Miller showed that $Q$-points are destroyed by iterating Laver (or Mathias) forcing, and Shelah produced a model without $P$-points.

Here's my (admittedly broad) question:

Let $\mathbb{P}$ be a nontrivial (say, infinite, separative) $\sigma$-closed notion of forcing which is in $L(\mathbb{R})$, by which I mean the underlying set, its elements, and its order are all in $L(\mathbb{R})$. Suppose that under CH one can define an ultrafilter $G$ in $\mathbb{P}$ which is generic over $L(\mathbb{R})$ (under suitable large cardinal hypothesis). Is there a general theorem which tells us that such a $G$ consistently does not exist?

  • 2
    $\begingroup$ I'd be surprised if the answer is positive to this broad case. Perhaps if you consider idealized forcings with "relatively definable" ideals (whatever that might be). $\endgroup$ – Asaf Karagila Apr 25 '16 at 10:41
  • $\begingroup$ I've been wondering about the version of this question for the partial order of countable partial $E_{0}$-selectors. I guess I'd expect there to be such a theorem in this case. $\endgroup$ – Paul Larson May 12 '16 at 1:50

The book draft linked to below shows that existence of a weakly compact Woodin cardinal implies the existence of $L(\mathbb{R})$-generic filters for the following partial orders (all ordered by containment) : (1) the partial order of countable injections from $\mathbb{R}$ to $\mathbb{R}$; (2) the partial order of countable partial selectors for a countable Borel equivalence relation; (3) the partial order of countable linearly independent sets in a Polish vector space over a countable field of scalars. The first example is Example 6.2.13 and the corresponding fact is Theorem 12.2.3. Examples (2) and (3) are discussed after Definition 6.2.9 (which they satisfy) and the corresponding theorem is Theorem 12.2.5. It should be possible to significantly increase the class of examples.

Here is the link : https://people.clas.ufl.edu/zapletal/files/balanced8.pdf


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