Axioms of generic absoluteness

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(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 $\mathcal{A}(W, Π_n , Γ)$

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

  • If $X^\sharp$ (sharp) exists for every set $X$, then $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2})$ holds.
  • $\mathcal{A}(H(ω_1), Σ_2)$ is equiconsistent with ZFC.
  • $\mathcal{A}(H(ω_1), \underset{\sim}{Σ_2})$ is equiconsistent with the existence of a $Σ_2$-reflecting cardinal.
  • ......
  • The existence of a $Σ_ω$-Mahlo cardinal is equiconsistent with $\mathcal{A}(L(\mathbb{R}), Σ_ω , Γ ∩ absolutely−ccc)$ 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.
  • ......
  • 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 $θ$.
  • ......

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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