# Difference between revisions of "Axioms of generic absoluteness"

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* $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2}, \textit{Hechler})$ 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. | * $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2}, \textit{Hechler})$ 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. | * $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2}, \textit{Amoeba})$ 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{ | + | * $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2}, \textit{Mathias})$ 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. | * $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2})$ is equivalent to the statement that every $\underset{\sim}{\Delta^1_2}$ set of reals is universally Baire. | ||

## Revision as of 14:36, 2 November 2019

(from [1]; 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.

## Contents

## 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 $κ > ω_1$
- $\mathcal{A}(V, Σ_1(H(ω_1)))$

- In particular, the following hold:
- $\mathcal{A}(V, \underset{\sim}{Σ_1}, \mathbb{P})$ 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 $V_α$, 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 $\mathcal{A}(H(κ), \underset{\sim}{Σ_1}, Γ)$.

- for regular $κ$ it is a model of ZFC without powerset and so it satisfies replacement.

### 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 $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2}, \textit{Semi-proper})$.
- $BSPFA$ implies $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2}, \textit{Stat-pres})$.
- The lest 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 $Σ_3$

- 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$-Random}, \textit{Cohen}\})$
- $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2}, \textit{Random}) \land \mathcal{A}(H(ω_1), Σ_3, \textit{Cohen})$

- 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 following is equiconsistent with the existence of a $Σ_2$-reflecting cardinal and a sharp for each set:
- $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_3})$

- ......

### ......

- The existence of a $Σ_ω$-Mahlo cardinal is equiconsistent with $\mathcal{A}(L(\mathbb{R}), Σ_ω , Γ ∩ \textit{absolutely−ccc})$ where $Γ$ is the class of 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?
- ......

## References

Main library