Difference between revisions of "Axioms of generic absoluteness"

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(Results for $H(ω_2)$ and $Σ_2$: full argument)
(Results for $H(ω_2)$ and $Σ_1$: correction, +2)
 
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(from <cite>Bagaria2002:AxiomsOfGenericAbsoluteness</cite>; compare [[Projective#Generically_absolute]])
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&#40;from <cite>Bagaria2002:AxiomsOfGenericAbsoluteness</cite>; 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
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'''Axioms of generic absoluteness''' are axioms $\mathcal{A}&#40;W, \Phi, \Gamma)$ of the form “$W$ is $\Phi$-elementarily equivalent to $W^{V^\mathbb{P}}$ for all $\mathbb{P} ∈ \Gamma$”, where
 
* $W$ is a subclass of $V$.
 
* $W$ is a subclass of $V$.
 
* $\Phi$ is a class of sentences.
 
* $\Phi$ is a class of sentences.
 
* $\Gamma$ is a class of [[forcing]] notions.
 
* $\Gamma$ 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 $\forall_{\phi\in\Phi} (W^V \models \phi \quad \text{iff} \quad W^{V^\mathbb{P}} \models \phi)$.
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* “$W^V$ is $\Phi$-elementarily equivalent to $W^{V^\mathbb{P}}$” &#40;symbolically $W^V \equiv_\Phi W^{V^\mathbb{P}}$) means that $\forall_{\phi\in\Phi} &#40;W^V \models \phi \quad \text{iff} \quad W^{V^\mathbb{P}} \models \phi)$.
$W$$, \Phi$ and $\Gamma$ must be definable classes for $\mathcal{A}(W, \Phi, \Gamma)$ to be a sentence in the first-order language of Set Theory.
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$W$$, \Phi$ and $\Gamma$ must be definable classes for $\mathcal{A}&#40;W, \Phi, \Gamma)$ to be a sentence in the first-order language of Set Theory.
  
 
== Notation ==
 
== Notation ==
* If $Γ$ contains only one element, $\mathbb{P}$, then one can write $\mathcal{A}(W, Φ, \mathbb{P})$ instead of $\mathcal{A}(W, Φ, Γ)$.
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* If $Γ$ contains only one element, $\mathbb{P}$, then one can write $\mathcal{A}&#40;W, Φ, \mathbb{P})$ instead of $\mathcal{A}&#40;W, Φ, Γ)$.
* If $Γ$ is the class of all set-forcing notions, then one can just write $\mathcal{A}(W, Φ)$.
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* If $Γ$ is the class of all set-forcing notions, then one can just write $\mathcal{A}&#40;W, Φ)$.
* The class of $\Sigma_n$ sentences with parameters from $W$ is denoted $\Sigma_n(W)$ or in short $\underset{\sim}{\Sigma_n}$.
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* The class of $\Sigma_n$ sentences with parameters from $W$ is denoted $\Sigma_n&#40;W)$ or in short $\underset{\sim}{\Sigma_n}$.
 
** Analogously for $\Pi_n$ etc.
 
** Analogously for $\Pi_n$ etc.
 
** Boldface $\mathbf{\Sigma_n}$ is used in other sources for similar notions.
 
** Boldface $\mathbf{\Sigma_n}$ is used in other sources for similar notions.
  
 
== Basic properties ==
 
== Basic properties ==
* If $Φ ⊆ Φ_0$ and $Γ ⊆ Γ_0$, then $\mathcal{A}(W, Φ_0 , Γ_0)$ implies $\mathcal{A}(W, Φ, Γ)$.
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* If $Φ ⊆ Φ_0$ and $Γ ⊆ Γ_0$, then $\mathcal{A}&#40;W, Φ_0 , Γ_0)$ implies $\mathcal{A}&#40;W, Φ, Γ)$.
* $\mathcal{A}(W, Φ, Γ)$ is equivalent to $\mathcal{A}(W, \bar{Φ}, Γ)$, where $\bar{Φ}$ is the closure of $Φ$ under finite Boolean combinations.
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* $\mathcal{A}&#40;W, Φ, Γ)$ is equivalent to $\mathcal{A}&#40;W, \bar{Φ}, Γ)$, where $\bar{Φ}$ is the closure of $Φ$ under finite Boolean combinations.
** Eg. $\mathcal{A}(W, Σ_n , Γ)$ is equivalent to $\mathcal{A}(W, Π_n , Γ)$
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** Eg. $\mathcal{A}&#40;W, Σ_n , Γ)$ is equivalent to $\mathcal{A}&#40;W, Π_n , Γ)$
* 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 $\mathbb{P} ∈ Γ$.
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* If $Φ ⊆ \underset{\sim}{Σ_0}$, then $\mathcal{A}&#40;W, Φ, Γ)$ 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 $\mathbb{P} ∈ Γ$.
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* If $Φ ⊆ Σ_1&#40;H&#40;ω_1))$, then &#40;by the Levy-Shoenfield absoluteness theorem) $\mathcal{A}&#40;W, Φ, Γ)$ 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:
 
** In particular, the following hold:
*** $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_1})$
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*** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_1})$
*** $\mathcal{A}(H(κ), Σ_1(H(ω_1)))$ for $κ > ω_1$
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*** $\mathcal{A}&#40;H&#40;κ), Σ_1&#40;H&#40;ω_1)))$ for $κ > ω_1$
*** $\mathcal{A}(V, Σ_1(H(ω_1)))$
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*** $\mathcal{A}&#40;V, Σ_1&#40;H&#40;ω_1)))$
* $\mathcal{A}(V, \underset{\sim}{Σ_1}, \mathbb{P})$ 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).
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* $\mathcal{A}&#40;V, \underset{\sim}{Σ_1}, \mathbb{P})$ fails for any nontrivial $\mathbb{P}$ &#40;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 $W = V$ &#40;contradiction).
* For every forcing notion $\mathbb{P}$, $L^V = L^{V^\mathbb{P}}$, so $A(L, Φ)$ holds for all $Φ$.
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* For every forcing notion $\mathbb{P}$, $L^V = L^{V^\mathbb{P}}$, so $A&#40;L, Φ)$ 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 $Φ$.
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* For every forcing notion $\mathbb{P}$, $H&#40;ω)^V = H&#40;ω)^{V^\mathbb{P}}$, so $A&#40;H&#40;ω), Φ)$ 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.
 
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 ==
 
== Results ==
Interesting results are obtained for $W = H(κ)$ or $W = L(H(κ))$ with some definable uncountable cardinal $κ$.
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Interesting results are obtained for $W = H&#40;κ)$ or $W = L&#40;H&#40;κ))$ with some definable uncountable cardinal $κ$.
* $H(κ)$ is better then $V_\alpha$ (for ordinal $\alpha$), because
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* $H&#40;κ)$ is better then $V_\alpha$ &#40;for ordinal $\alpha$), because
 
** for regular $κ$ it is a model of ZFC without powerset and so it satisfies replacement.
 
** 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 $H(κ)$.
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*** This allows for nice properties like: if $\mathbb{P} ∈ H&#40;κ)$, then a filter $G ⊆ \mathbb{P}$ is generic over $V$ iff it is generic over $H&#40;κ)$.
** If $κ < λ$, then $\mathcal{A}(H(λ), \underset{\sim}{Σ_1}, Γ)$ implies $\mathcal{A}(H(κ), \underset{\sim}{Σ_1}, Γ)$.
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** If $κ < λ$, then $\mathcal{A}&#40;H&#40;λ), \underset{\sim}{Σ_1}, Γ)$ implies $\mathcal{A}&#40;H&#40;κ), \underset{\sim}{Σ_1}, Γ)$.
  
=== Results for $H(ω_1)$ and $Σ_2$ ===
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=== Results for $H&#40;ω_1)$ and $Σ_2$ ===
 
Relations with large cardinal properties:
 
Relations with large cardinal properties:
* If [[zero sharp|$X^\sharp$]] ([[zero sharp|sharp]]) exists for every set $X$, then $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2})$ holds.
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* If [[zero sharp|$X^\sharp$]] &#40;[[zero sharp|sharp]]) exists for every set $X$, then $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2})$ holds.
 
* The following are equiconsistent with ZFC:
 
* The following are equiconsistent with ZFC:
** $\mathcal{A}(H(ω_1), Σ_2)$
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** $\mathcal{A}&#40;H&#40;ω_1), Σ_2)$
** $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2}, \textit{Semi-proper})$
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** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Semi-proper})$
* $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2}, \textit{Semi-proper})$ does not imply that $ω_1^L$ is countable.
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* $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Semi-proper})$ 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 $L$.
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* If $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Proper})$ 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:
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* The following are equiconsistent with the existence of a $Σ_2$-[[reflecting cardinals|reflecting]] cardinal:
** $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2})$
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** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2})$
** $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2}, \textit{Stat-pres})$
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** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Stat-pres})$
*** (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 $x$.)
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*** &#40;Because $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Stat-pres})$ implies that $ω_1$ is a $Σ_2$-reflecting cardinal in $L[x]$ for every real $x$.)
  
 
Relations with bounded forcing axioms:
 
Relations with bounded forcing axioms:
* $MA_{ω_1}$ implies $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2}, \textit{ccc})$.
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* $MA_{ω_1}$ implies $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{ccc})$.
* $BPFA$ implies $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2}, \textit{Proper})$.
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* $BPFA$ implies $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Proper})$.
* $BSPFA$ implies $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2}, \textit{Semi-proper})$.
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* $BSPFA$ implies $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Semi-proper})$.
* $BMM$ implies $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2}, \textit{Stat-pres})$.
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* $BMM$ implies $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Stat-pres})$.
* 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.
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* The last four implications cannot be reversed, because all axioms of the form $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_n}, \textit{Stat-pres})$ 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
 
* 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:
 
** $ω_1$ 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 $θ$.
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** $MA_{ω_1}$ is equivalent to $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{ccc})$ plus $θ$.
** $BPFA$ is equivalent to $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2}, \textit{Proper})$ plus $θ$.
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** $BPFA$ is equivalent to $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Proper})$ plus $θ$.
** $BSPFA$ is equivalent to $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2}, \textit{Semi-proper})$ plus $θ$.
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** $BSPFA$ is equivalent to $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Semi-proper})$ plus $θ$.
 
* $BSPFA$ is consistent with $ω_1^L = ω_1$.
 
* $BSPFA$ is consistent with $ω_1^L = ω_1$.
 
* $BMM$ implies that $ω_1$ is weakly compact in $L[x]$ for every $x ⊆ ω$.
 
* $BMM$ implies that $ω_1$ is weakly compact in $L[x]$ for every $x ⊆ ω$.
  
 
Equivalences to other statements:
 
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.
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* $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Hechler})$ and $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Amoeba-category})$ 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.
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* $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Amoeba})$ 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.
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* $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Mathias})$ 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 [[Projective#Suslin_sets_and_universally_Baire_sets|universally Baire]].
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* $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2})$ is equivalent to the statement that every $\underset{\sim}{\Delta^1_2}$ set of reals is [[Projective#Suslin_sets_and_universally_Baire_sets|universally Baire]].
  
=== Results for $H(ω_1)$ and $Σ_3$ ===
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=== Results for $H&#40;ω_1)$ and $Σ_3$ ===
 
Relations with large cardinal properties:
 
Relations with large cardinal properties:
 
* Each of the following implies that $ω_1$ is inaccessible in $L[x]$ for every real $x$:
 
* 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$-Random}, \textit{Cohen}\})$
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** $\mathcal{A}&#40;H&#40;ω_1), Σ_3, \{\textit{$ω_1$-Random}, \textit{Cohen}\})$
** $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2}, \textit{Random}) \land \mathcal{A}(H(ω_1), Σ_3, \textit{Cohen})$
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** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_2}, \textit{Random}) \land \mathcal{A}&#40;H&#40;ω_1), Σ_3, \textit{Cohen})$
** $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_3}, \textit{Mathias})$
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** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_3}, \textit{Mathias})$
** $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_3}, \textit{Hechler})$
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** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_3}, \textit{Hechler})$
 
* The following are equiconsistent with the existence of a sharp for each set:
 
* The following are equiconsistent with the existence of a sharp for each set:
** $\mathcal{A}(H(ω_1), Σ_3)$
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** $\mathcal{A}&#40;H&#40;ω_1), Σ_3)$
** $\mathcal{A}(H(ω_1), Σ_3, \textit{Stat-pres})$
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** $\mathcal{A}&#40;H&#40;ω_1), Σ_3, \textit{Stat-pres})$
** $\mathcal{A}(H(ω_1), Σ_3, \textit{$ω_1$-pres})$ (obviously from the other two)
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** $\mathcal{A}&#40;H&#40;ω_1), Σ_3, \textit{$ω_1$-pres})$ &#40;obviously from the other two)
* The existence of a $Σ_2$-reflecting cardinal and a sharp for each set is equiconsistent with $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_3})$.
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* The existence of a $Σ_2$-reflecting cardinal and a sharp for each set is equiconsistent with $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_3})$.
 
* The following are equiconsistent with the existence of a weakly compact cardinal for $3 \le n \le \omega$:
 
* The following are equiconsistent with the existence of a weakly compact cardinal for $3 \le n \le \omega$:
** $\mathcal{A}(H(ω_1), Σ_n, \textit{Knaster})$
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** $\mathcal{A}&#40;H&#40;ω_1), Σ_n, \textit{Knaster})$
** $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_n}, \textit{ccc})$
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** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_n}, \textit{ccc})$
 
* The following are equiconsistent with the existence of a Mahlo cardinal for $3 \le n \le \omega$:
 
* 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$-centered})$
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** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_n}, \textit{$\sigma$-centered})$
** $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_n}, \textit{$\sigma$-linked})$
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** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_n}, \textit{$\sigma$-linked})$
 
* The following are equiconsistent with the existence of an inaccessible cardinal for $3 \le n \le \omega$:
 
* 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$-Random}, \textit{Cohen}\})$
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** $\mathcal{A}&#40;H&#40;ω_1), Σ_n, \{\textit{$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-$\underset{\sim}{Σ_2}$
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** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_n}, \Gamma)$ 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
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** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_n}, \Gamma)$ where $\Gamma$ is the class of strongly-proper posets that are $Σ_2$ definable in $H&#40;ω_1)$ 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 $L$.)
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** &#40;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 $H&#40;ω_1)$, without parameters, and for which the axiom $\mathcal{A}&#40;H&#40;ω_1), Σ_3, \mathbb{P})$ 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 $3 \le n \le \omega$:
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* The following are equiconsistent with the existence of a [[remarkable]] cardinal for $3 \le n \le \omega$:
** $\mathcal{A}(H(ω_1), Σ_n, \textit{Proper})$
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** $\mathcal{A}&#40;H&#40;ω_1), Σ_n, \textit{Proper})$
** $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_n}, \textit{Proper})$
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** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_n}, \textit{Proper})$
  
 
Relations with bounded forcing axioms:
 
Relations with bounded forcing axioms:
 
* If $x^\sharp$ exists for every real $x$ and the second uniform indiscernible is $< ω_2$, then
 
* If $x^\sharp$ exists for every real $x$ and the second uniform indiscernible is $< ω_2$, then
** $MA_{ω_1}$ implies $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_3}, \textit{ccc})$.
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** $MA_{ω_1}$ implies $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_3}, \textit{ccc})$.
** $BPFA$ implies $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_3}, \textit{Proper})$.
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** $BPFA$ implies $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_3}, \textit{Proper})$.
** $BSPFA$ implies $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_3}, \textit{Semi-proper})$.
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** $BSPFA$ implies $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_3}, \textit{Semi-proper})$.
** $BMM$ implies $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_3}, \textit{Stat-pres})$.
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** $BMM$ implies $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_3}, \textit{Stat-pres})$.
  
 
Relations with other statements:
 
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.
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* $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_3}, \{\textit{Amoeba-category}, \textit{Cohen}\})$ 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.
+
* $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_3}, \{\textit{Amoeba}, \textit{Random}\})$ 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.
+
* $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_3}, \textit{Mathias})$ 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.
+
* $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_3})$ implies that every $\underset{\sim}{\Delta^1_3}$ set of reals is universally Baire.
 
** The converse does not hold.
 
** The converse does not hold.
  
=== Results for $H(ω_1)$ and $Σ_n$, $4 \le n \le ω$ ===
+
=== Results for $H&#40;ω_1)$ 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 $X$.
+
* $\mathcal{A}&#40;H&#40;ω_1), Σ_4)$ implies that [[zero dagger|$X^\dagger$ &#40;dagger)]] exists for every set $X$.
 
* The following are equiconsistent with the existence of infinitely many [[strong]] cardinals:
 
* The following are equiconsistent with the existence of infinitely many [[strong]] cardinals:
** $\mathcal{A}(H(ω_1), Σ_ω)$
+
** $\mathcal{A}&#40;H&#40;ω_1), Σ_ω)$
** $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_ω})$
+
** $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_ω})$
* If there is a proper class of [[Woodin]] cardinals, then $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_ω})$.
+
* If there is a proper class of [[Woodin]] cardinals, then $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_ω})$.
* 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 existence of a $Σ_ω$-Mahlo cardinal is equiconsistent with $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_ω}, Γ)$ 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 of a $Σ_ω$-Mahlo cardinal implies consistency of $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_ω}, Γ)$ 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
+
* The consistency strength of $\mathcal{A}&#40;H&#40;ω_1), \underset{\sim}{Σ_n})$ for $n \ge 4$ is
 
** at least that of $n-3$ strong cardinals
 
** 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.
 
** 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 $H&#40;ω_2)$ and $Σ_1$ ===
 +
Relations with large cardinal properties:
 +
* The following are equiconsistent with the existence of a $Σ_2$-reflecting cardinal:
 +
** $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_1}, \textit{Proper})$
 +
** $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_1}, \textit{Semi-proper})$
 +
 
 +
Equivalence to bounded forcing axioms:
 +
* $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_1}, \textit{ccc}) \iff MA_{\omega_1}$
 +
** So $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_1}, \textit{ccc})$ is consistent with ZFC, because Martin's axiom is consistent with ZFC.
 +
** More generally: For any ccc poset $\mathbb{P}$, $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_1}, \mathbb{P}) \iff MA_{\omega_1}&#40;\mathbb{P})$
 +
* $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_1}, \textit{Proper}) \iff BPFA$
 +
* $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_1}, \textit{Semi-proper}) \iff BSPFA$
 +
* $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_1}, \textit{Stat-pres}) \iff BMM$
 +
 
 +
Other:
 +
* $\mathcal{A}&#40;H&#40;ω_2), Σ_1)$ holds &#40;as most cases with $Φ ⊆ Σ_1&#40;H&#40;ω_1))$ do, see section "Basic properties").
 +
* $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_1}, \mathbb{P})$ implies $\neg CH$ for any $\mathbb{P}$ that adds a real number.
 +
* $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_1}, \textit{Amoeba})$ is equivalent to the $ω_1$-additivity of the Lebesgue measure.
 +
* $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_1}, \textit{Amoeba-category})$ is equivalent to the $ω_1$-additivity of the Baire property.
 +
* $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_1}, \mathbb{P})$ is inconsistent with ZFC for any $\mathbb{P}$ that collapses $ω_1$.
 +
* $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_1}, ω_1\textit{-pres})$ is inconsistent with ZFC.
 
* ......
 
* ......
  
=== Results for $H(ω_2)$ and $Σ_2$ ===
+
=== Results for $H&#40;ω_2)$ and $Σ_2$ ===
* $\mathcal{A}(H(ω_2), Σ_2, \textit{$\sigma$-centered}) \land \neg CH$ is false.
+
* $\mathcal{A}&#40;H&#40;ω_2), Σ_2, \textit{$\sigma$-centered}) \land \neg CH$ is false.
 
** Because:
 
** Because:
*** by adding $ω_1$ Cohen reals (a $σ$-centered forcing notion) one adds a Luzin set (an uncountable set of reals that intersects every meager set in at most a countable set; its existence is a $Σ_2$ statement in $H(ω_2)$)
+
*** by adding $ω_1$ Cohen reals &#40;a $σ$-centered forcing notion) one adds a Luzin set &#40;an uncountable set of reals that intersects every meager set in at most a countable set; its existence is a $Σ_2$ statement in $H&#40;ω_2)$)
*** and then we may iterate in length the continuum $\textit{Amoeba-category}$ (another $σ$-centered forcing notion), so that in the generic extension every set of size $ω_1$ is meager.
+
*** and then we may iterate in length the continuum $\textit{Amoeba-category}$ &#40;another $σ$-centered forcing notion), so that in the generic extension every set of size $ω_1$ is meager.
* $\mathcal{A}(H(ω_2), Σ_2, \textit{Knaster})$ is false.
+
* $\mathcal{A}&#40;H&#40;ω_2), Σ_2, \textit{Knaster})$ is false.
 
** The above argument applies because any iteration of $\textit{Amoeba-category}$ with finite support is Knaster.
 
** 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.
+
* $\mathcal{A}&#40;H&#40;ω_2), \underset{\sim}{Σ_2}, \textit{$\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.
+
** The argument applies because given any set of reals in $H&#40;ω_2)$ we can force with $\textit{Amoeba-category}$ to make it meager.
  
=== Results for $H(\kappa)$, $\kappa \ge \omega_3$ ===
+
=== Results for $H&#40;\kappa)$, $\kappa \ge \omega_3$ ===
 
* ......
 
* ......
  
=== Results for $L(H(ω_1))$ ($=L(\mathbb{R})$) ===
+
=== Results for $L&#40;H&#40;ω_1))$ &#40;$=L&#40;\mathbb{R})$) ===
* The consistency strength of $\mathcal{A}(L(\mathbb{R}), Σ_ω(\mathbb{R}))$ is roughly that of the existence of infinitely many Woodin cardinals:
+
&#40;$L&#40;H&#40;ω_1))=L&#40;\mathbb{R})$, because every element of $H&#40;ω_1)$ can be easily coded by a real number.)
** 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 $L(H(ω_2))$ ===
+
Results:
 +
* The consistency strength of $\mathcal{A}&#40;L&#40;\mathbb{R}), Σ_ω&#40;\mathbb{R}))$ is roughly that of the existence of infinitely many Woodin cardinals:
 +
** If there is a proper class of Woodin cardinals, then $\mathcal{A}&#40;L&#40;\mathbb{R}), Σ_ω&#40;\mathbb{R}))$ holds.
 +
** $\mathcal{A}&#40;L&#40;\mathbb{R}), Σ_ω&#40;\mathbb{R}))$ implies that the [[axiom of determinacy]] holds in $L&#40;\mathbb{R})$ &#40;$\mathrm{AD}^{L&#40;\mathbb{R})}$, equiconsistent with $\mathrm{AD}$, equiconsistent with the existence of infinitely many Woodin cardinals).
 +
* If $δ$ is a weakly compact Woodin cardinal, then $\mathcal{A}&#40;L&#40;\mathbb{R}), \underset{\sim}{Σ_ω}, \mathbb{P})$ holds for every proper poset $\mathbb{P} ∈ V_δ$.
 +
** Therefore, $\mathcal{A}&#40;L&#40;\mathbb{R}), \underset{\sim}{Σ_ω}, \textit{Proper})$ follows from the existence of a proper class of weakly compact Woodin cardinals.
 +
** The existence of just a remarkable cardinal is equiconsistent with $\mathcal{A}&#40;L&#40;\mathbb{R}), \underset{\sim}{Σ_ω}, \textit{Proper})$.
 +
* The following are equiconsistent with the existence of a weakly compact cardinal:
 +
** $\mathcal{A}&#40;L&#40;\mathbb{R}), \underset{\sim}{Σ_ω}, \textit{Knaster})$
 +
** $\mathcal{A}&#40;L&#40;\mathbb{R}), \underset{\sim}{Σ_ω}, \textit{ccc})$
 +
* The following are equiconsistent with the existence of a Mahlo cardinal:
 +
** $\mathcal{A}&#40;L&#40;\mathbb{R}), \underset{\sim}{Σ_ω}, \textit{$\sigma$-centered})$
 +
** $\mathcal{A}&#40;L&#40;\mathbb{R}), \underset{\sim}{Σ_ω}, \textit{$\sigma$-linked})$
 +
* The existence of a $Σ_ω$-Mahlo cardinal is equiconsistent with $\mathcal{A}&#40;L&#40;\mathbb{R}), \underset{\sim}{Σ_ω}, Γ)$ where $Γ$ is the class of absolutely-ccc projective posets.
 +
* The consistency of a $Σ_ω$-Mahlo cardinal implies consistency of $\mathcal{A}&#40;L&#40;\mathbb{R}), \underset{\sim}{Σ_ω}, Γ)$ where $Γ$ is the class of strongly proper projective posets.
 +
* The existence of a $Σ_n$-weakly compact cardinal is equiconsistent with $\mathcal{A}&#40;L&#40;\mathbb{R}), \underset{\sim}{Σ_ω}, Γ_n)$ where $Γ_n$ is the class of ccc posets that are $Σ_n$ or $Π_n$ definable in $H&#40;ω_1)$ with parameters.
 +
* The existence of a $Σ_ω$-weakly compact cardinal is equiconsistent with $\mathcal{A}&#40;L&#40;\mathbb{R}), \underset{\sim}{Σ_ω}, Γ)$ where $Γ$ is the class of projective ccc forcing notions.
 +
 
 +
=== Results for $L&#40;H&#40;ω_2))$ ===
 
* ......
 
* ......
  
 
== Open problems ==
 
== 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?
+
* Does $\mathcal{A}&#40;H&#40;ω_1), Σ_ω , Γ)$, for $Γ$ the class of Borel ccc forcing notions, imply that every [[projective]] set of real numbers is Lebesgue measurable?
 
* ......
 
* ......
  
 
{{References}}
 
{{References}}
 +
 +
[[Category:Large cardinal axioms]]
 +
[[Category:Forcing]]

Latest revision as of 13:15, 14 May 2022

(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.

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.

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 $Σ_1$

Relations with large cardinal properties:

  • The following are equiconsistent with the existence of a $Σ_2$-reflecting cardinal:
    • $\mathcal{A}(H(ω_2), \underset{\sim}{Σ_1}, \textit{Proper})$
    • $\mathcal{A}(H(ω_2), \underset{\sim}{Σ_1}, \textit{Semi-proper})$

Equivalence to bounded forcing axioms:

  • $\mathcal{A}(H(ω_2), \underset{\sim}{Σ_1}, \textit{ccc}) \iff MA_{\omega_1}$
    • So $\mathcal{A}(H(ω_2), \underset{\sim}{Σ_1}, \textit{ccc})$ is consistent with ZFC, because Martin's axiom is consistent with ZFC.
    • More generally: For any ccc poset $\mathbb{P}$, $\mathcal{A}(H(ω_2), \underset{\sim}{Σ_1}, \mathbb{P}) \iff MA_{\omega_1}(\mathbb{P})$
  • $\mathcal{A}(H(ω_2), \underset{\sim}{Σ_1}, \textit{Proper}) \iff BPFA$
  • $\mathcal{A}(H(ω_2), \underset{\sim}{Σ_1}, \textit{Semi-proper}) \iff BSPFA$
  • $\mathcal{A}(H(ω_2), \underset{\sim}{Σ_1}, \textit{Stat-pres}) \iff BMM$

Other:

  • $\mathcal{A}(H(ω_2), Σ_1)$ holds (as most cases with $Φ ⊆ Σ_1(H(ω_1))$ do, see section "Basic properties").
  • $\mathcal{A}(H(ω_2), \underset{\sim}{Σ_1}, \mathbb{P})$ implies $\neg CH$ for any $\mathbb{P}$ that adds a real number.
  • $\mathcal{A}(H(ω_2), \underset{\sim}{Σ_1}, \textit{Amoeba})$ is equivalent to the $ω_1$-additivity of the Lebesgue measure.
  • $\mathcal{A}(H(ω_2), \underset{\sim}{Σ_1}, \textit{Amoeba-category})$ is equivalent to the $ω_1$-additivity of the Baire property.
  • $\mathcal{A}(H(ω_2), \underset{\sim}{Σ_1}, \mathbb{P})$ is inconsistent with ZFC for any $\mathbb{P}$ that collapses $ω_1$.
  • $\mathcal{A}(H(ω_2), \underset{\sim}{Σ_1}, ω_1\textit{-pres})$ is inconsistent with ZFC.
  • ......

Results for $H(ω_2)$ and $Σ_2$

  • $\mathcal{A}(H(ω_2), Σ_2, \textit{$\sigma$-centered}) \land \neg CH$ is false.
    • Because:
      • by adding $ω_1$ Cohen reals (a $σ$-centered forcing notion) one adds a Luzin set (an uncountable set of reals that intersects every meager set in at most a countable set; its existence is a $Σ_2$ statement in $H(ω_2)$)
      • and then we may iterate in length the continuum $\textit{Amoeba-category}$ (another $σ$-centered forcing notion), so that in the generic extension every set of size $ω_1$ is meager.
  • $\mathcal{A}(H(ω_2), Σ_2, \textit{Knaster})$ 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 $L(H(ω_1))$ ($=L(\mathbb{R})$)

($L(H(ω_1))=L(\mathbb{R})$, because every element of $H(ω_1)$ can be easily coded by a real number.)

Results:

  • 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).
  • If $δ$ is a weakly compact Woodin cardinal, then $\mathcal{A}(L(\mathbb{R}), \underset{\sim}{Σ_ω}, \mathbb{P})$ holds for every proper poset $\mathbb{P} ∈ V_δ$.
    • Therefore, $\mathcal{A}(L(\mathbb{R}), \underset{\sim}{Σ_ω}, \textit{Proper})$ follows from the existence of a proper class of weakly compact Woodin cardinals.
    • The existence of just a remarkable cardinal is equiconsistent with $\mathcal{A}(L(\mathbb{R}), \underset{\sim}{Σ_ω}, \textit{Proper})$.
  • The following are equiconsistent with the existence of a weakly compact cardinal:
    • $\mathcal{A}(L(\mathbb{R}), \underset{\sim}{Σ_ω}, \textit{Knaster})$
    • $\mathcal{A}(L(\mathbb{R}), \underset{\sim}{Σ_ω}, \textit{ccc})$
  • The following are equiconsistent with the existence of a Mahlo cardinal:
    • $\mathcal{A}(L(\mathbb{R}), \underset{\sim}{Σ_ω}, \textit{$\sigma$-centered})$
    • $\mathcal{A}(L(\mathbb{R}), \underset{\sim}{Σ_ω}, \textit{$\sigma$-linked})$
  • 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 consistency of a $Σ_ω$-Mahlo cardinal implies consistency of $\mathcal{A}(L(\mathbb{R}), \underset{\sim}{Σ_ω}, Γ)$ where $Γ$ is the class of strongly proper projective posets.
  • The existence of a $Σ_n$-weakly compact cardinal is equiconsistent with $\mathcal{A}(L(\mathbb{R}), \underset{\sim}{Σ_ω}, Γ_n)$ where $Γ_n$ is the class of ccc posets that are $Σ_n$ or $Π_n$ definable in $H(ω_1)$ with parameters.
  • 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

  1. Bagaria, Joan. Axioms of generic absoluteness. Logic Colloquium 2002 , 2006. www   DOI   bibtex
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