# Difference between revisions of "Axioms of generic absoluteness"

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=== Results for $H(ω_1)$ and $Σ_n$, $4 \le n \le ω$ === | === Results for $H(ω_1)$ and $Σ_n$, $4 \le n \le ω$ === | ||

+ | * $\mathcal{A}(H(ω_1), Σ_4)$ implies that [[zero dagger|$X^\dagger$ (dagger)]] exists for every set $X$. | ||

+ | * The following are equiconsistent with the existence of infinitely many [[strong]] cardinals: | ||

+ | ** $\mathcal{A}(H(ω_1), Σ_ω)$ | ||

+ | ** $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_ω})$ | ||

+ | * If there is a proper class of [[Woodin]] cardinals, then $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_ω})$. | ||

+ | * The existence of a $Σ_ω$-Mahlo cardinal is equiconsistent with $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_ω}, Γ)$ where $Γ$ is the class of absolutely-ccc projective posets. | ||

+ | * The consistency of a $Σ_ω$-Mahlo cardinal implies consistency of $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_ω}, Γ)$ where $Γ$ is the class of strongly proper projective posets. | ||

+ | * The consistency strength of $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_n})$ for $n \ge 4$ is | ||

+ | ** at least that of $n-3$ strong cardinals | ||

+ | ** and at most that of $n-3$ strong cardinals with a $Σ_2$-reflecting cardinal above them. | ||

+ | |||

+ | === Results for $H(ω_2)$ and $Σ_1$ === | ||

* ...... | * ...... | ||

− | === ...... === | + | === Results for $H(ω_2)$ and $Σ_2$ === |

+ | * $\mathcal{A}(H(ω_2), Σ_2, \textit{$\sigma$-centered}) \land \neg CH$ is false. | ||

+ | ** Because of a Luzin set (an uncountable set of reals that intersects every meager set in at most a countable set) and $\textit{Amoeba-category}$ ...... | ||

+ | * $\mathcal{A}(H(ω_2), Σ_2, \textit{Knaster})$ is false. | ||

+ | ** The above argument applies because any iteration of $\textit{Amoeba-category}$ with finite support is Knaster. | ||

+ | * $\mathcal{A}(H(ω_2), \underset{\sim}{Σ_2}, \textit{$\sigma$-centered})$ is false. | ||

+ | ** The argument applies because given any set of reals in $H(ω_2)$ we can force with $\textit{Amoeba-category}$ to make it meager. | ||

+ | |||

+ | === Results for $H(\kappa)$, $\kappa \ge \omega_3$ === | ||

+ | * ...... | ||

+ | |||

+ | === Results for $L(H(ω_1))$ ($=L(\mathbb{R})$) === | ||

* The consistency strength of $\mathcal{A}(L(\mathbb{R}), Σ_ω(\mathbb{R}))$ is roughly that of the existence of infinitely many Woodin cardinals: | * 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 | + | ** 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). | ** $\mathcal{A}(L(\mathbb{R}), Σ_ω(\mathbb{R}))$ 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 $Σ_ω$-Mahlo cardinal is equiconsistent with $\mathcal{A}(L(\mathbb{R}), \underset{\sim}{Σ_ω}, Γ)$ 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. | + | * 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. |

+ | * ...... | ||

+ | |||

+ | === Results for $L(H(ω_2))$ === | ||

* ...... | * ...... | ||

## Revision as of 01:20, 5 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

- 1 Notation
- 2 Basic properties
- 3 Results
- 3.1 Results for $H(ω_1)$ and $Σ_2$
- 3.2 Results for $H(ω_1)$ and $Σ_3$
- 3.3 Results for $H(ω_1)$ and $Σ_n$, $4 \le n \le ω$
- 3.4 Results for $H(ω_2)$ and $Σ_1$
- 3.5 Results for $H(ω_2)$ and $Σ_2$
- 3.6 Results for $H(\kappa)$, $\kappa \ge \omega_3$
- 3.7 Results for $L(H(ω_1))$ ($=L(\mathbb{R})$)
- 3.8 Results for $L(H(ω_2))$

- 4 Open problems
- 5 References

## 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 $V_\alpha$ (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 $\mathcal{A}(H(κ), \underset{\sim}{Σ_1}, Γ)$.

- 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 $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2}, \textit{Semi-proper})$.
- $BMM$ 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.

### Results for $H(ω_1)$ and $Σ_n$, $4 \le n \le ω$

- $\mathcal{A}(H(ω_1), Σ_4)$ implies that $X^\dagger$ (dagger) exists for every set $X$.
- The following are equiconsistent with the existence of infinitely many strong cardinals:
- $\mathcal{A}(H(ω_1), Σ_ω)$
- $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_ω})$

- If there is a proper class of Woodin cardinals, then $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_ω})$.
- The existence of a $Σ_ω$-Mahlo cardinal is equiconsistent with $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_ω}, Γ)$ where $Γ$ is the class of absolutely-ccc projective posets.
- The consistency of a $Σ_ω$-Mahlo cardinal implies consistency of $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_ω}, Γ)$ where $Γ$ is the class of strongly proper projective posets.
- The consistency strength of $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_n})$ for $n \ge 4$ is
- at least that of $n-3$ strong cardinals
- and at most that of $n-3$ strong cardinals with a $Σ_2$-reflecting cardinal above them.

### Results for $H(ω_2)$ and $Σ_1$

- ......

### Results for $H(ω_2)$ and $Σ_2$

- $\mathcal{A}(H(ω_2), Σ_2, \textit{$\sigma$-centered}) \land \neg CH$ is false.
- Because of a Luzin set (an uncountable set of reals that intersects every meager set in at most a countable set) and $\textit{Amoeba-category}$ ......

- $\mathcal{A}(H(ω_2), Σ_2, \textit{Knaster})$ is false.
- The above argument applies because any iteration of $\textit{Amoeba-category}$ with finite support is Knaster.

- $\mathcal{A}(H(ω_2), \underset{\sim}{Σ_2}, \textit{$\sigma$-centered})$ is false.
- The argument applies because given any set of reals in $H(ω_2)$ we can force with $\textit{Amoeba-category}$ to make it meager.

### Results for $H(\kappa)$, $\kappa \ge \omega_3$

- ......

### Results for $L(H(ω_1))$ ($=L(\mathbb{R})$)

- 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 $\mathcal{A}(L(\mathbb{R}), \underset{\sim}{Σ_ω}, Γ)$ 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.
- ......

### Results for $L(H(ω_2))$

- ......

## 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