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.

4 thoughts on “Oracle forcing Part I

  1. Ok, I am going to stop using markdown now in wordpress. All and even sometime quotes are changed *in the html editor* (!) to their respective ASCII codes as soon as I press update. This didn’t happen last summer! The visual editor is disabled, etc.. I cannot use blockquotes as these are changed before the markdown processor can get to it. Back to good old html for now…

    • Those markdown troubles are really odd. It sounds more like a bug somewhere to me and I’d love to hunt it down — I never had the experience you describe :(

      I also wondered if the first definition is missing some text (like “oracle”). Also, the first sentence after M-cc is defined seems to be missing some text.

  2. Aren’t you in fact the young queen of set theory? This is much the most interesting
    blog post I have read this week, can I have the sooper sekrit password to
    the sequel or is that reserved for younger set theorists than me?

  3. James: flattery will get you far, but sending the promised correspondence would get you even farther. ;)

    Peter: thanks for noticing the missing words (which also happened as some of the markdown turned html simply disappeared) – I will put those in anon.

    Part II is forthcoming as public – there are 1 or 2 proofs missing that I thought I had somewhere….

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