Characterizing forcings that don't add any dominating reals - MathOverflow

Characterizing forcings that don't add any dominating reals

2010-11-20T19:54:24Z

Regarding reals as functions from $\omega$ to $\omega$, let's say a real $f$ eventually dominates $g$ iff $(\exists n)(\forall m > n)[ f(m) > g(m)]$. Let's say that a (non-trivial separative) forcing poset $P$ doesn't always add a dominating real iff there is a generic extension by $P$ which doesn't contains a real that eventually dominates every real from the ground model. Let's say that $P$ never adds a dominating real iff every generic extension by $P$ doesn't contain any real that eventually dominates all the ground model's reals. I'm interested in combinatorial/order-theoretic conditions which may be necessary or sufficient for either of these notions.

$\omega$-closure implies you add no reals, hence you add no dominating reals; Cohen forcing is not $\omega$-closed but it never adds a dominating real
one can show that separability implies you never add a dominating real (by separability, I mean containing a countable dense subset); the Cohen forcing that adds uncountably many reals isn't separable but never adds a dominating real
Hechler forcing has size at most continuum but always adds a dominating real; the Cohen forcing that adds more than continuum many reals (where "continuum" is "continuum as computed in the ground model" obviously) has size greater than continuum but never adds a dominating real
Hechler forcing also has the countable chain condition yet adds a dominating real; the forcing that adds a function $\omega _1 \to \omega _1$ with countable partial functions doesn't have the ccc but it's $\omega$-closed hence adds no new reals and thus never adds any dominating reals.

My question:

What are some combinatorial/order-theoretic conditions on a poset that are necessary and/or sufficient for the poset to never/not always add a dominating real?

Answer by Stefan Geschke for Characterizing forcings that don't add any dominating reals

2010-11-20T22:05:02Z

Let me first say something about not always adding a dominating real (or anything else that can be formalized in the forcing language).

Given a forcing notion $P$, the Forcing Theorem (everthing true in the extension is forced by an element of the filter) implies that there is a dense set $D$ of conditions that either force that there is a dominating real in the extension or that there is no dominating real in the extension Let $A$ be a maximal antichain of conditions in $D$. By the density of $D$, $A$ is a maximal antichain in $P$. Hence every generic filter will pick exactly one condition $p\in A$ and we can restrict our attention to the forcing notion that consists of conditions below $p$.

In other words, forcing notions that sometimes but not always add a dominating real are characterized by: there is a condition below which the forcing always adds a dominating real and there is a condition below which the forcing does not add a dominating real.

So, we are left with characterizing forcing notions that never add dominating reals respectively always add a dominating real.
I am sure it is possible (and maybe a good exercise) to write down a characterization of adding or not adding a dominating real, but I am not aware of any crisp, helpful combinatorial characterization of either of these properties. I might be missing something, though.
Natural places to check are the Bartoszynski-Judah book Set Theory of the Real Line and Zapletal's Forcing Idealized.

Something that is well understood, though, is $\omega^\omega$-boundingness. A forcing notion is $\omega^\omega$-bounding if every function from $\omega$ to $\omega$ in the extension is bounded by a function in the ground model.
This turns out to be a form of distributivity, so called weak $(\omega,\omega)$-distributivity (this is covered in Jech's Set Theory, The Third Millenium Edition and I would guess in earlier editions as well). Clearly, $(\omega,\omega)$-bounding forcings do not add dominating reals.
Examples are random real forcing (forcing with Borel sets of the real line of positive measure) and Sacks forcing.

Answer by Haim for Characterizing forcings that don't add any dominating reals

2010-11-21T16:59:36Z

There is a somewhat related result by Shelah: Any Suslin ccc forcing which adds a non-dominated real adds a Cohen real. The proof can be found here: http://shelah.logic.at/files/480.pdf

Answer by Amit Kumar Gupta for Characterizing forcings that don't add any dominating reals

2010-11-23T04:33:52Z

Stefan's answer pointed me in the right direction, and then talking it over with prof. Leo Harrington we've got an answer:

A complete Boolean algebra $\mathbb{B}$ never adds a dominating real iff for any collection $\{ u _{m,k} : m, k \in \omega \} \subset \mathbb{B}^+$ the following weaker form of weak $(\omega ,\omega )$-distributivity holds:

$\prod _{m \in \omega} \sum _{k \in \omega} u _{m,k} = \sum _{f \in \omega ^{\omega}} \prod _{n \in \omega} \sum _{n < m < \omega} \sum _{k < f(m)} u _{m,k}$

For the reverse implication, suppose "weak weak $(\omega ,\omega )$-distributivity" holds, and for contradiction let $u \neq 0$ be the Boolean value of the sentence "there exists a dominating real," and let $\dot{g}$ be a name witnessing this, i.e. the sentence "$\dot{g}$ is a dominating real" has Boolean value $u$. Define $u _{m,k} = || \dot{g} (m) = k ||$. Now if $G$ is any $\mathbb{B}$-generic filter containing $u$, noting that the left side of the distributivity identity is (at least) $u$, we know that the right side belongs to $G$. It's then not hard to see that:

$(\exists f \in (\omega ^{\omega})^V )(\forall n \in \omega )(\exists m > n)(\exists k < f(m))(u _{m,k} \in G)$

which is to say that there's a real $f$ in the ground model such that:

$(\forall n)(\exists m > n)(\dot{g}^G (m) < f(m))$

so $f$ is not dominated by $\dot{g}$, contradiction.

For the forward implication, it should suffice to show it in the case where for each $m$, the set $\{ u _{m,k} : k \in \omega \}$ is an antichain with least upper bound $u$ independent of $m$ (I haven't checked this detail personally). So let $\{ u _{m,k}\}$ be such a collection for which the identity fails. Consider the name:

$\dot{g} = \{ (u _{m,k}, (m,k)) : m, k \in \omega \}$

It's not hard to see that the right side of the identity is at most $u$, so assuming the identity fails it's strictly less than $u$, so since $\mathbb{B}$ is separative there's a generic $G$ containing $u$ avoiding the right side of the identity. It's not hard to see from here that $\dot{g}^G$ will dominate all the ground model's reals.

I should add that if we think of $u _{m,n}$ as saying "$\dot{g}(m) = n$" and replace the Boolean operations with the corresponding quantifiers, then the left side says "$\dot{g}$ is a real," and the right side says "$\dot{g}$ doesn't dominate every real in the ground model." This suggests how we can characterize forcings that don't add any unbounded reals, for example, namely the following identity holds:

$\prod _{m \in \omega} \sum _{k \in \omega} u _{m,k} = \sum _{f \in \omega ^{\omega}} \prod _{m \in \omega} \sum _{k < f(m)} u _{m,k}$

Forcings that don't add any reals are precisely those that satisfy the following identity:

$\prod _{m \in \omega} \sum _{k \in \omega} u _{m,k} = \sum _{f \in \omega ^{\omega}} \prod _{m \in \omega} u _{m,f(m)}$

You can easily generalize this to talking about functions $\kappa \to \lambda$; the above two results so generalized are precisely Theorem 15.38 and Lemma 15.39 in Jech, "Set Theory".