Forcing Preservation on Arbitrary Set Theoretic Formulas

Fixing a ground model $M$, a forcing notion $\mathbb{P}$ is called cardinal preserving iff for all $\mathbb{P}$-generic filter $G$ over $M$ we have:

$$\forall a\in M~~~(M\models card(a)\Longleftrightarrow M[G]\models card(a))$$

which $card(x)$ is a $\{\in\}$-formula asserting "$x$ is a cardinal".

Also a forcing notion $\mathbb{P}$ is called cofinality preserving iff for all $\mathbb{P}$-generic filter $G$ over $M$ we have:

$$\forall a,b\in M~~~(M\models cof(a,b)\Longleftrightarrow M[G]\models cof(a,b))$$

which $cof(x,y)$ is a $\{\in\}$-formula asserting "$x$,$y$ are ordinals and $x$ is limit and cofinality of $x$ is $y$".

It is well-known that each cofinality preserving forcing is also cardinal preserving. Also based on a construction of Prikry, there are examples of forcing notions which are cardinal preserving and don't preserve cofinalities.

Let us generalize the forcing preservation notion to arbitrary formulas as follows:

Definition 1: Fixing a ground model $M$ and a $\{\in\}$-formula $\varphi(x_1,...,x_n)$ a forcing notion $\mathbb{P}$ is called $\varphi$-preserving iff for all $\mathbb{P}$-generic filter $G$ over $M$ we have:

$$\forall a_1,..,a_n\in M~~~(M\models \varphi(a_1,...,a_n)\Longleftrightarrow M[G]\models \varphi(a_1,...,a_n))$$

Definition 2: Fixing a ground model $M$ and a set $T$ of $\{\in\}$-formulas, a forcing notion $\mathbb{P}$ is called $T$-preserving iff for all $\varphi\in T$, the forcing notion $\mathbb{P}$ is $\varphi$-preserving.

Now define the "forcing preservation order" as follows:

Definition 3: Fixing a ground model $M$ and two $\{\in\}$-formulas $\varphi(x_1,...,x_n)$ and $\psi(x_1,...,x_m)$ we say $\varphi$ is weaker than $\psi$ in forcing preservation order ($\varphi\leq_{F.P}\psi$) iff for all forcing notions $\mathbb{P}$ we have:

$$\mathbb{P}~~\text{preserves}~~\psi\Longrightarrow \mathbb{P}~~\text{preserves}~~\varphi$$

Notation 1: $\varphi\sim_{F.P}\psi \Longleftrightarrow \varphi\leq_{F.P}\psi~\wedge~\psi\leq_{F.P}\varphi$ ($\sim_{F.P}$ is an equivalence relation on the set of $\{\in\}$-formulas). For sets of $\{\in\}$-formulas define the notation similarly.

Notation 2: $\varphi<_{F.P}\psi \Longleftrightarrow \varphi\leq_{F.P}\psi~\wedge~\varphi\nsim_{F.P}\psi$. For sets of $\{\in\}$-formulas define the notation similarly.

Example: $card(x)<_{F.P} cof(x,y)$ and so $\{card(x)\}<_{F.P} \{cof(x,y)\}$.

Question 1: Let $\{\in\}-Form$ denotes the set of all $\{\in\}$-formulas. Is it true that $|\frac{\{\in\}-Form}{\sim_{F.P}}|=\aleph_0$ and $|\frac{P(\{\in\}-Form)}{\sim_{F.P}}|=2^{\aleph_0}$?

Question 2: Fixing a ground model $M$ and an arbitrary $\{\in\}$-formula $\varphi(x_1,...,x_n)$, is there a forcing notion $\mathbb{P}$ such that:

(a) There exists $\mathbb{P}$-generic filter $G$ over $M$ such that $M\nprec M[G]$.

(b) $\mathbb{P}$ is a $\varphi$-preserving forcing notion.

• Note that if $M[G]\neq M$ then $M\nprec M[G]$. This is true since $M$ is definable in $M[G]$ (with parameters from $M$), and the set of generic filters for $M$ is definable in $M[G]$ (after defining $M$, of course); so by elementarity we would have that the set of generic filters is in $M$. That cannot be. – Asaf Karagila Jun 1 '14 at 19:54
• I'm confused. Why isn't Question 3 just trivial by taking unions of everything preserved? – Miha Habič Jun 1 '14 at 21:26
• It might not have much to do with the actual answer, but it still worth pointing out, I think, that $\Sigma_1$ formulas cannot be changed by forcing, as they are upwards absolute between a model and its generic extensions. – Asaf Karagila Jun 1 '14 at 22:30
• @AsafKaragila Yes, but the OP defined preservation as going both ways (which is a bit confusing to me). – Miha Habič Jun 1 '14 at 22:51
• @Miha: So $\Delta_1$ formulas. – Asaf Karagila Jun 1 '14 at 22:53

For the first part of Question 1, the answer is yes. This can be seen by considering the sentences $\sigma_n$ saying that the continuum equals $\aleph_n$. Assuming for convenience that GCH holds in $M$, all of these are pairwise nonequivalent since, if $n<m$, the forcing $\operatorname{Add}(\omega,\aleph_n)$ preserves $\sigma_m$ but not $\sigma_n$. A similar approach works for part 2, where to each function $f\colon\omega\to\omega$ you assign the set of sentences $T_f=\{\psi_{n,f(n)};n<\omega\}$ where $\psi_{n,m}$ says that $2^{\aleph_n}=\aleph_{\omega\cdot n + m+1}$.
For Question 2 the answer is no, since $M$ might have certain special properties. As Asaf alludes to in the comments, the ground model is always definable in a forcing extension, from a ground model parameter (this is surprisingly fairly recent, due to Laver, Hamkins and Woodin). It was later noticed by Reitz that the definition is completely uniform. It follows that there is a sentence $\sigma$ which holds in $M$ iff $M$ is not a (nontrivial set) forcing extension (Reitz calls this the ground axiom). Clearly if this $\sigma$ holds in the ground model, it can never be preserved by nontrivial set forcing.