# Axioms of generic absoluteness

(from ; compare Projective#Generically_absolute)

Axioms of generic absoluteness are axioms $\mathcal{A}(W, \Phi, \Gamma)$ of the form “$W$ is $\Phi$-elementarily equivalent to $W^{V^\mathbb{P}}$ for all $\mathbb{P} ∈ \Gamma$”, where

• $W$ is a subclass of $V$.
• $\Phi$ is a class of sentences.
• $\Gamma$ is a class of forcing notions.
• “$W^V$ is $\Phi$-elementarily equivalent to $W^{V^\mathbb{P}}$” (symbolically $W^V \equiv_\Phi W^{V^\mathbb{P}}$) means that $\forall_{\phi\in\Phi} (W^V \models \phi \quad \text{iff} \quad W^{V^\mathbb{P}} \models \phi)$.

$W$$, \Phi$ and $\Gamma$ must be definable classes for $\mathcal{A}(W, \Phi, \Gamma)$ to be a sentence in the first-order language of Set Theory.

## Notation

• If $Γ$ contains only one element, $\mathbb{P}$, then one can write $\mathcal{A}(W, Φ, \mathbb{P})$ instead of $\mathcal{A}(W, Φ, Γ)$.
• If $Γ$ is the class of all set-forcing notions, then one can just write $\mathcal{A}(W, Φ)$.
• The class of $\Sigma_n$ sentences with parameters from $W$ is denoted $\Sigma_n(W)$ or in short $\underset{\sim}{\Sigma_n}$.
• Analogously for $\Pi_n$ etc.
• Boldface $\mathbf{\Sigma_n}$ is used in other sources for similar notions.

## Basic properties

• If $Φ ⊆ Φ_0$ and $Γ ⊆ Γ_0$, then $\mathcal{A}(W, Φ_0 , Γ_0)$ implies $\mathcal{A}(W, Φ, Γ)$.
• $\mathcal{A}(W, Φ, Γ)$ is equivalent to $\mathcal{A}(W, \bar{Φ}, Γ)$, where $\bar{Φ}$ is the closure of $Φ$ under finite Boolean combinations.
• Eg. $\mathcal{A}(W, Σ_n , Γ)$ is equivalent to $\mathcal{A}(W, Π_n , Γ)$
• If $Φ ⊆ \underset{\sim}{Σ_0}$, then $\mathcal{A}(W, Φ, Γ)$ holds for all transitive $W$ and all $Γ$ such that $W^V$ is contained in $W^{V^\mathbb{P}}$ for all $\mathbb{P} ∈ Γ$.
• If $Φ ⊆ Σ_1(H(ω_1))$, then (by the Levy-Shoenfield absoluteness theorem) $\mathcal{A}(W, Φ, Γ)$ holds for every transitive model $W$ of a weak fragment of ZF that contains the parameters of $Φ$, and all $Γ$, provided $W^V$ is contained in $W^{V^\mathbb{P}}$ for all $\mathbb{P} ∈ Γ$.
• In particular, the following hold:
• $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_1})$
• $\mathcal{A}(H(κ), Σ_1(H(ω_1)))$ for $κ > ω_1$
• $\mathcal{A}(V, Σ_1(H(ω_1)))$
• $\mathcal{A}(V, \underset{\sim}{Σ_1}, \mathbb{P})$ fails for any nontrivial $\mathbb{P}$ (any $\mathbb{P}$ that adds some new set), because a proper class $W \neq V$ can never be an elementary substructure of $V$, since otherwise, by elementarity, $V_α^W = V_α$ for every ordinal $α$ and so $W = V$ (contradiction).
• For every forcing notion $\mathbb{P}$, $L^V = L^{V^\mathbb{P}}$, so $A(L, Φ)$ holds for all $Φ$.
• For every forcing notion $\mathbb{P}$, $H(ω)^V = H(ω)^{V^\mathbb{P}}$, so $A(H(ω), Φ)$ holds for all $Φ$.

We see that, when $W = V$, $\Phi$ is the class of all sentences or $\Gamma$ is the class of all forcing notions, then the other two must be very small for the axiom to be consistent with ZFC.

## Results

Interesting results are obtained for $W = H(κ)$ or $W = L(H(κ))$ with some definable uncountable cardinal $κ$.

• $H(κ)$ is better then $V_\alpha$ (for ordinal $\alpha$), because
• for regular $κ$ it is a model of ZFC without powerset and so it satisfies replacement.
• This allows for nice properties like: if $\mathbb{P} ∈ H(κ)$, then a filter $G ⊆ \mathbb{P}$ is generic over $V$ iff it is generic over $H(κ)$.
• If $κ < λ$, then $\mathcal{A}(H(λ), \underset{\sim}{Σ_1}, Γ)$ implies $\mathcal{A}(H(κ), \underset{\sim}{Σ_1}, Γ)$.

### Results for $H(ω_1)$ and $Σ_2$

Relations with large cardinal properties:

• If $X^\sharp$ (sharp) exists for every set $X$, then $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2})$ holds.
• The following are equiconsistent with ZFC:
• $\mathcal{A}(H(ω_1), Σ_2)$
• $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2}, \textit{Semi-proper})$
• $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2}, \textit{Semi-proper})$ does not imply that $ω_1^L$ is countable.
• If $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2}, \textit{Proper})$ holds after forcing with a certain proper poset, then either $ω_1$ is Mahlo in $L$ or $ω_2$ is inaccessible in $L$.
• The following are equiconsistent with the existence of a $Σ_2$-reflecting cardinal:
• $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2})$
• $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2}, \textit{Stat-pres})$
• (Because $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2}, \textit{Stat-pres})$ implies that $ω_1$ is a $Σ_2$-reflecting cardinal in $L[x]$ for every real $x$.)

Relations with bounded forcing axioms:

• $MA_{ω_1}$ implies $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2}, \textit{ccc})$.
• $BPFA$ implies $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2}, \textit{Proper})$.
• $BSPFA$ implies $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2}, \textit{Semi-proper})$.
• $BMM$ implies $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2}, \textit{Stat-pres})$.
• The last four implications cannot be reversed, because all axioms of the form $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_n}, \textit{Stat-pres})$ are preserved after collapsing the continuum to $ω_1$ by $σ$-closed forcing and so are all consistent with CH and do not imply any of the bounded forcing axioms.
• If $θ$ is the statement that every subset of $ω_1$ is constructible from a real, that is, for every $X ⊆ ω_1$ there is $x ⊆ ω$ with $X ∈ L[x]$ and
• $ω_1$ is not weakly compact in $L[x]$ for some $x ⊆ ω$, then:
• $MA_{ω_1}$ is equivalent to $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2}, \textit{ccc})$ plus $θ$.
• $BPFA$ is equivalent to $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2}, \textit{Proper})$ plus $θ$.
• $BSPFA$ is equivalent to $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2}, \textit{Semi-proper})$ plus $θ$.
• $BSPFA$ is consistent with $ω_1^L = ω_1$.
• $BMM$ implies that $ω_1$ is weakly compact in $L[x]$ for every $x ⊆ ω$.

Equivalences to other statements:

• $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2}, \textit{Hechler})$ and $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2}, \textit{Amoeba-category})$ are both equivalent to the statement that every $\underset{\sim}{Σ^1_2}$ set of reals has the property of Baire.
• $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2}, \textit{Amoeba})$ is equivalent to the statement that every $\underset{\sim}{Σ^1_2}$ set of reals is Lebesgue measurable.
• $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2}, \textit{Mathias})$ is equivalent to the statement that every $\underset{\sim}{Σ^1_2}$ set of reals is Ramsey.
• $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2})$ is equivalent to the statement that every $\underset{\sim}{\Delta^1_2}$ set of reals is universally Baire.

### Results for $H(ω_1)$ and $Σ_3$

Relations with large cardinal properties:

• Each of the following implies that $ω_1$ is inaccessible in $L[x]$ for every real $x$:
• $\mathcal{A}(H(ω_1), Σ_3, \{\textit{$ω_1$-Random}, \textit{Cohen}\})$
• $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2}, \textit{Random}) \land \mathcal{A}(H(ω_1), Σ_3, \textit{Cohen})$
• $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_3}, \textit{Mathias})$
• $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_3}, \textit{Hechler})$
• The following are equiconsistent with the existence of a sharp for each set:
• $\mathcal{A}(H(ω_1), Σ_3)$
• $\mathcal{A}(H(ω_1), Σ_3, \textit{Stat-pres})$
• $\mathcal{A}(H(ω_1), Σ_3, \textit{$ω_1$-pres})$ (obviously from the other two)
• The existence of a $Σ_2$-reflecting cardinal and a sharp for each set is equiconsistent with $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_3})$.
• The following are equiconsistent with the existence of a weakly compact cardinal for $3 \le n \le \omega$:
• $\mathcal{A}(H(ω_1), Σ_n, \textit{Knaster})$
• $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_n}, \textit{ccc})$
• The following are equiconsistent with the existence of a Mahlo cardinal for $3 \le n \le \omega$:
• $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_n}, \textit{$\sigma$-centered})$
• $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_n}, \textit{$\sigma$-linked})$
• The following are equiconsistent with the existence of an inaccessible cardinal for $3 \le n \le \omega$:
• $\mathcal{A}(H(ω_1), Σ_n, \{\textit{$ω_1$-Random}, \textit{Cohen}\})$
• $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_n}, \Gamma)$ where $\Gamma$ is the class of posets that are absolutely-ccc and strongly-$\underset{\sim}{Σ_2}$
• $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_n}, \Gamma)$ where $\Gamma$ is the class of strongly-proper posets that are $Σ_2$ definable in $H(ω_1)$ with parameters
• (This result is optimal, for there is a, provably in ZFC, ccc poset $\mathbb{P}$ which is both $Σ_2$ and $Π_2$ definable in $H(ω_1)$, without parameters, and for which the axiom $\mathcal{A}(H(ω_1), Σ_3, \mathbb{P})$ fails if $ω_1$ is not a $Π_1$-Mahlo cardinal in $L$.)
• The following are equiconsistent with the existence of a remarkable cardinal for $3 \le n \le \omega$:
• $\mathcal{A}(H(ω_1), Σ_n, \textit{Proper})$
• $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_n}, \textit{Proper})$

Relations with bounded forcing axioms:

• If $x^\sharp$ exists for every real $x$ and the second uniform indiscernible is $< ω_2$, then
• $MA_{ω_1}$ implies $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_3}, \textit{ccc})$.
• $BPFA$ implies $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_3}, \textit{Proper})$.
• $BSPFA$ implies $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_3}, \textit{Semi-proper})$.
• $BMM$ implies $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_3}, \textit{Stat-pres})$.

Relations with other statements:

• $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_3}, \{\textit{Amoeba-category}, \textit{Cohen}\})$ implies that every $\underset{\sim}{\Delta^1_3}$ set of reals has the property of Baire.
• $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_3}, \{\textit{Amoeba}, \textit{Random}\})$ implies that every $\underset{\sim}{\Delta^1_3}$ set of reals is Lebesgue measurable.
• $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_3}, \textit{Mathias})$ implies that every $\underset{\sim}{\Sigma^1_3}$ set of reals is Ramsey.
• $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_3})$ implies that every $\underset{\sim}{\Delta^1_3}$ set of reals is universally Baire.
• The converse does not hold.

• ......

### ......

• The consistency strength of $\mathcal{A}(L(\mathbb{R}), Σ_ω(\mathbb{R}))$ is roughly that of the existence of infinitely many Woodin cardinals:
• If there is a proper class of Woodin cardinals, then $\mathcal{A}(L(\mathbb{R}), Σ_ω(\mathbb{R}))$ holds.
• $\mathcal{A}(L(\mathbb{R}), Σ_ω(\mathbb{R}))$ implies that the axiom of determinacy holds in $L(\mathbb{R})$ ($\mathrm{AD}^{L(\mathbb{R})}$, equiconsistent with $\mathrm{AD}$, equiconsistent with the existence of infinitely many Woodin cardinals).
• The existence of a $Σ_ω$-Mahlo cardinal is equiconsistent with $\mathcal{A}(L(\mathbb{R}), Σ_ω , Γ)$ where $Γ$ is the class of absolutely-ccc projective posets.
• The existence of a $Σ_ω$-weakly compact cardinal is equiconsistent with $\mathcal{A}(L(\mathbb{R}), \underset{\sim}{Σ_ω} , Γ)$ where $Γ$ is the class of projective ccc forcing notions.
• ......

## Open problems

• Does $\mathcal{A}(H(ω_1), Σ_ω , Γ)$, for $Γ$ the class of Borel ccc forcing notions, imply that every projective set of real numbers is Lebesgue measurable?
• ......