Oracle forcing part II

Putting oracle forcing into context

$\sigma$-closed
$\Downarrow$
uberoracle-proper $\Rightarrow$ strong $\bar{M}$-proper $\Longrightarrow$
$\Rightarrow$ $\bar{M}$-proper
proper
$\Uparrow$ $\Uparrow$ $\Uparrow$ $\Uparrow$
uberoracle-cc $\Rightarrow$ strong $\bar{M}$-cc $\Rightarrow$ $\bar{M}$-cc $\Rightarrow$ ccc
$\Uparrow$
$\sigma$-centred

Note: All arrows are strict. The only arrow which could possibly be added is $\sigma$-centred $\Rightarrow \bar{M}$-proper; this is unknown. The picture is supposed to show that strong oracle-proper implies proper, but oracle-proper does not imply proper.

Proofs of these implications and counterexamples to show strictness are given below (at least the ones that are not obvious, well-known or mentioned in the previous post), working our way from left to right in the picture.

Claim Any $\sigma$-closed forcing is $\bar{M}$-proper for any oracle $\bar{M}$.

Proof Starting with a condition $p$ and $\delta \in \omega_1$, basically use the closure of the forcing to extend $p$ to a $(\delta, M_\delta)$-generic condition.
$\Box$

To see that uberoracle-cc implies uberoracle-proper and that strong oracle-cc implies strong oracle-proper, one simply uses the fact that if a forcing is $\bar{M}$-cc for a specific oracle $\bar{M}$ then it is $\bar{M}$-proper for the same oracle (see Abraham’s notes for a proof of this latter fact).

The following claim (proof due to Martin is forthcoming) implies both that the notion of uberoracle-proper is strictly stronger than oracle-proper and that uberoracle-cc is stronger than oracle-cc.

Claim:

  1. For any oracle $\bar{M}$ there is a Suslin tree $T$ such that $T$ is $\bar{M}$-cc.
  2. For any Suslin tree $T$ and any oracle $\bar{M}$ there is $\bar{M}^\prime \trianglerighteq \bar{M}$ such that $T$ is not $\bar{M}^\prime$-proper.

The next claim can be found in Shelah 100.

Claim For any strong oracle $\bar{M}$, if $P$ is strong $\bar{M}$-proper and $|P| \leq \aleph_1$ then $P$ is proper.

Proof Assume $P = \omega_1$. Fix $p \in P$. Let $N$ be a countable elementary submodel of some large $H(\xi)$ such that $\bar{M}, P \in N$. Let $\delta^* = N \cap \omega_1$.

Let $S$ be the set of all $\delta < \omega_1$ such that there exists $q \leq p$ where $q$ is $(\delta, M_\delta)$-generic which is club as $P$ is $\bar{M}$-proper and $\bar{M}$ is a strong oracle. Therefore we may assume that that $\delta^* \in S$.

Now let $E \in N$ such that $E \subseteq P$ and dense in $P$. The set $C$ of $\delta \in \omega_1$ such that $E \cap P \upharpoonright \delta$ is dense in $P\upharpoonright \delta$ is club and $C \in N$ so $\delta^* \in C$. Also $E \cap P \upharpoonright \delta^* \in N$. We want to see that $E \cap P\upharpoonright \delta^* \in M_{\delta^*}$.

The set
$$\{N : N \cap [\omega_1]^{\aleph_0} \subseteq M_{(N \cap \omega_1)}\}$$
is club in $[H(\xi)]^{\aleph_0}$.
Why? Given $N_i : i < \omega$ in this set, we have
$$(\bigcup N_i) \cap [\omega_1]^{\aleph_0} \subseteq \bigcup_i M_{(N_i \cap \omega_1)} \subseteq M_{(\bigcup N_i \cap \omega_1)}$$
so it is closed. Thus the $N$ fixed above is a member of this set, which gives
$$E \cap P\upharpoonright \delta^* \in M_{\delta^*}.$$

Now by $\bar{M}$-properness we have $E \cap P\upharpoonright \delta^* = E \cap N$ is predense in $P$ below $q$.

$\Box$

This next one though, proved by Martin, comes as somewhat a surprise and makes us think twice about bothering at all with the weak definition of oracle-proper.

Claim: There is a $P$ which is $\bar{M}$-proper and $|P| = \aleph_1$ but $P$ is not proper.

Proof Let $S$ be a stationary, co-stationary subset of $\omega_1$ and let $P$ be the forcing which collapses $\omega_1 \setminus S$. That is, conditions are continuous functions $f : \alpha \rightarrow \alpha$ where $\alpha$ is a successor ordinal and $f(\beta) \in S$ for all $\beta < \alpha$. This forcing is not proper, as models $N$ such that $N \cap \omega_1 \not\in S$ do not have $(N,P)$-generic conditions.

Let $\bar{M}$ be defined as $\{M_\delta : \delta \in S\}$. Given $\delta \in S$ and $p \in M_\delta$ we will find $q \leq p$ which is $(\delta, M_\delta)$-generic. Denote by $P \upharpoonright \delta = \{f \in P : f \subset \delta \times \delta\}$. Enumerate by $\langle A_n : n < \omega\rangle$ the set of $A \subseteq P\upharpoonright \delta$ such that $D \in M_\delta$ and are antichains in $P\upharpoonright \delta$. We may extend $p$ in $\omega$-steps such that each $p_n$ forces that the generic intersects $A_n$ at a point $a_n$. Then $q = \bigcup p_n \cup \{(\delta, \delta)\}$ is a condition in $P$ and forces that for all $n < \omega$ the $P$-generic filter meets $A_n$ at $a_n$ (i.e. is non-empty).
$\Box$

Finally we show that the oracle-cc to oracle-proper implications are strict.

Claim: There is a forcing which is uberoracle-proper but neither $\omega$-proper nor ccc.

Proof Let conditions in $P$ be finite partial functions $p : \omega_1 \rightarrow \omega_1$ which are weakly increasing. This is proper, but not Axiom A, see Jech Ch. 31 exercises.

To see that this forcing is oracle proper for any oracle, let $p \in P$ and $\delta$ be such that $P \upharpoonright \delta = \{p \upharpoonright (\delta \times \delta) : p \in P\}$ (happens on a club). Then for $M_\delta$ we let $q = p \cup \{\delta, \delta\}$ which is $(\delta, M_\delta)$-generic.
$\Box$

Oracle forcing Part I

Introduction and definitions

Recently, Martin Goldstern and I have been studying oracle-cc and oracle-proper forcing, which was introduced by Shelah in his paper numbered 100 (proper forcing was also introduced in this paper) which is creatively and descriptively titled Independence Results.

Before I get to what these things are, a brief note on why. Oracle forcings in all variants have the property that the ground model reals remain of second category in the extension. They also have the Omitting Types Theorem or “what is forced to be dead, stays dead” (details can be found in Proper and Improper Forcing, Chapter IV).

There are some applications of oracle-cc forcing (e.g. Geschke-Kojman or Mildenberger) but so far the only application of oracle-proper forcing is the one given in the original Shelah 100 paper. Shelah uses an iteration of oracle-proper forcings to come up with a model in which there is a universal linear order at $\aleph_1$ and CH fails. This application is important, the method of getting the universal is deceptively simple and so the technique is worth studying.

Martin and I started by studying Uri Abrahams notes from a tutorial given at the Logic Colloquium in Paris in 2010. The definitions and the first conceptual ideas that I will state stem from those notes, but are ultimately the same as the original Shelah concepts. Then we went on to study how these notions of oracle forcing fit into the framework of familiar forcing categories such as proper, ccc, $\sigma$-closed, $\sigma$-centred. I will try to relate this work here.

Definition: A sequence $\bar{M} = \langle M_\delta : \delta < \omega_1,$ limit $\rangle$ is an oracle (strong oracle) iff

  • for each $\delta$ $M_\delta$ is a countable elementary submodel of some $H(\xi)$ for sufficiently large $\xi$ such that $\delta \in M_\delta$ and $M_\delta \vDash \delta$ is countable

  • for any $A \subseteq \omega_1$ the set of all $\delta \in \omega_1$ such that $A \cap \delta \in M_{\delta}$ is stationary (club).

To see that the existence of oracles is equivalent to $\diamondsuit_{\aleph_1}$, first let $$\bar{A} = \langle A_{\alpha} : \alpha \in \omega_1\rangle$$
be a $\diamondsuit_{\aleph_1}$-sequence and for each limit $\delta \in \omega_1$ let $M_{\delta}$ be the collapse of a countable elementary subset of $H(\xi)$ containing $A_\delta$ (for some $\xi$ sufficiently large) such that $\delta \in M_{\delta}$.

The opposite direction is shown by taking as $A_\delta$ all subsets of $\delta$ in $M_\delta$ (and $A_\delta = \emptyset$ for $\delta$ a successor ordinal). This is a easily seen to be a $\diamondsuit^-$ sequence which is equivalent to $\diamondsuit$ by a theorem of Kunen.

Definition: An oracle (strong oracle) $\bar{M}^\prime = \langle M^\prime_{\delta} : \delta \in \omega_1, \text{ limit}\rangle$ is a proper extension of an oracle (strong oracle) $\bar{M}$, denoted $\bar{M} \trianglelefteq \bar{M^\prime}$, if for a club subset of $\delta \in \omega_1$

  • $M_{\delta} \in M^\prime_{\delta}$ and $M_{\delta} \subseteq M^\prime_{\delta}$

  • $M^\prime_{\delta} \vDash |M_{\delta}| = \aleph_0$.

Shelah proves that given any oracle, we may find another oracle which is a proper extension of it. In fact, this process may be repeated $\aleph_1$-many times.

For an oracle $\bar{M}$, and $A \subseteq \omega_1$ let $I_{\bar{M}}(A)$ be the set of non-zero limit ordinals $\delta \in \omega_1$ such that $A \cap \delta \in M_\delta$. Let $D_{\bar{M}}$ be the filter which is generated by $I_{\bar{M}}(A)$ for all $A \subseteq \omega_1$. The filter $D_{\bar{M}}$ is normal and contains the club filter; proofs of these facts can be found in Abrahams notes.

Definition: Let $\bar{M}$ be an oracle and let $P$ be a forcing poset such that the universe of $P$ is $\omega_1$. Let $D(P)$ be the set of all $\delta \in \omega_1$ such that for all $E \in M_\delta$, $E$ is pre-dense in $P \upharpoonright \delta$ implies $E$ is pre-dense in $P$. Then $P$ is $\bar{M}$-cc iff $D(P) \in D_{\bar{M}}$.

A simpler way of writing this is that $D(P) = \{\delta \in \omega_1 : P_\delta \preceq_{M_\delta} P\} \in D_{\bar{M}}$.

The fact that any $\bar{M}$-cc satisfies the ccc can be found in Abrahams notes.

Definition: Let $\bar{M}$ be an oracle and let $P$ be a forcing poset such that the universe of $P$ is $\omega_1$. For any $p \in P$ let $D_p(P)$ be the set of all $\delta \in \omega_1$ such that there exists $q_\delta \leq p$ for all $E \in M_\delta$, $E$ is pre-dense in $P \upharpoonright \delta$ implies $E$ is pre-dense in $P$ below $q_\delta$. Then $P$ is $\bar{M}$-proper iff $D_p(P) \in D_{\bar{M}}$.

Such $q_\delta$ are called $(\delta, M_\delta)$-generic.

As Abraham points out, we can combine the sets $D_p(P)$ for $p \in P$ to obtain a single set $D(P)$ such that if $\delta \in D(P)$ for each $p \in P \upharpoonright \delta$ there is $q \leq p$ which is $(\delta, M_\delta)$-generic.

In order to work with $P$ which do not have universe $\omega_1$ Abraham provides an equivalent definition. Let $K \prec H(\lambda)$ have size $\aleph_1$ such that $P \in K$, and let $s_\alpha : K_\alpha \rightarrow \bar{K_\alpha}$ be the collapse function. Denote $\bar{P}_\alpha = s_\alpha(P)$, the copy of $P$ in $\bar{K_\alpha}$. Let $\delta \in \omega_1$ be such that $\bar{K_\alpha} \in M_\delta$. We say that $q \in P$ is $(K_\alpha, M_\delta)$-generic if for every $X \in M_\delta$ dense in $\bar{P}_\alpha$ we have $s^{-1}(X) \subseteq P \cap K_\alpha$ is predense in $P$ below $q$.

Theorem (Abraham):

$P$ is $\bar{M}$-proper iff for every $K$ and $\langle K_\alpha : \alpha \in \omega_1 \rangle$ as above, there is $I(P) \in D_{\bar{M}}$ such that for each $\delta \in I(P)$ with $\bar{K_\delta} \in M_\delta$ and for every $p \in P \cap K_\delta$ there is $q \leq p$ which is $(K_\delta, M_\delta)$-generic.

Notation: We say that a forcing is uberoracle-cc (uberoracle-proper) if and only if it is $\bar{M}$-cc ($\bar{M}$-proper) for every oracle $\bar{M}$. Also, we abbreviate the property “$\bar{M}$-cc ($\bar{M}$-proper) for a strong oracle $\bar{M}$” by strong $\bar{M}$-cc (strong $\bar{M}$-proper).

The reason that we define uberoracle-proper is that the only known application of oracle-proper forcing has this stronger property.