Uplifting cardinals
Uplifting cardinals were introduced by Hamkins and Johnstone in [1], from which some of this text is adapted.
An inaccessible cardinal $\kappa$ is uplifting if and only if for every ordinal $\theta$ it is $\theta$-uplifting, meaning that there is an inaccessible $\gamma>\theta$ such that $V_\kappa\prec V_\gamma$ is a proper elementary extension.
An inaccessible cardinal is pseudo uplifting if and only if for every ordinal $\theta$ it is pseudo $\theta$-uplifting, meaning that there is a cardinal $\gamma>\theta$ such that $V_\kappa\prec V_\gamma$ is a proper elementary extension, without insisting that $\gamma$ is inaccessible.
An inaccessible cardinal is strongly uplifting if and only for every ordinal $\theta$ it is strongly $\theta$-uplifting, meaning that for every $A\subseteq V_\kappa$, there exists some inaccessible $\lambda>\theta$ and an $A^*\subseteq V_\lambda$ such that $(V_\kappa;\in,A)\prec(V_\lambda;\in,A*)$ is a proper elementary extension. Equivalently, $\kappa$ is superstrongly unfoldable.
It is an elementary exercise to see that if $V_\kappa\prec V_\gamma$ is a proper elementary extension, then $\kappa$ and hence also $\gamma$ are $\beth$-fixed points, and so $V_\kappa=H_\kappa$ and $V_\gamma=H_\gamma$. It follows that a cardinal $\kappa$ is uplifting if and only if it is regular and there are arbitrarily large regular cardinals $\gamma$ such that $H_\kappa\prec H_\gamma$. It is also easy to see that every uplifting cardinal $\kappa$ is uplifting in $L$, with the same targets. Namely, if $V_\kappa\prec V_\gamma$, then we may simply restrict to the constructible sets to obtain $V_\kappa^L=L^{V_\kappa}\prec L^{V_\gamma}=V_\gamma^L$. An analogous result holds for pseudo-uplifting cardinals.
Contents
Consistency strength of uplifting cardinals
The consistency strength of uplifting and pseudo-uplifting cardinals are bounded between the existence of a Mahlo cardinal and the hypothesis Ord is Mahlo.
Theorem.
1. If $\delta$ is a Mahlo cardinal, then $V_\delta$ has a proper class of uplifting cardinals.
2. Every uplifting cardinal is pseudo uplifting and a limit of pseudo uplifting cardinals.
3. If there is a pseudo uplifting cardinal, or indeed, merely a pseudo $0$-uplifting cardinal, then there is a transitive set model of ZFC with a reflecting cardinal and consequently also a transitive model of ZFC plus Ord is Mahlo.
Proof. For (1), suppose that $\delta$ is a Mahlo cardinal. By the Lowenheim-Skolem theorem, there is a club set $C\subset\delta$ of cardinals $\beta$ with $V_\beta\prec V_\delta$. Since $\delta$ is Mahlo, the club $C$ contains unboundedly many inaccessible cardinals. If $\kappa<\gamma$ are both in $C$, then $V_\kappa\prec V_\gamma$, as desired. Similarly, for (2), if $\kappa$ is uplifting, then $\kappa$ is pseudo uplifting and if $V_\kappa\prec V_\gamma$ with $\gamma$ inaccessible, then there are unboundedly many ordinals $\beta<\gamma$ with $V_\beta\prec V_\gamma$ and hence $V_\kappa\prec V_\beta$. So $\kappa$ is pseudo uplifting in $V_\gamma$. From this, it follows that there must be unboundedly many pseudo uplifting cardinals below $\kappa$. For (3), if $\kappa$ is inaccessible and $V_\kappa\prec V_\gamma$, then $V_\gamma$ is a transitive set model of ZFC in which $\kappa$ is reflecting, and it is thus also a model of Ord is Mahlo. QED
Uplifting cardinals and $\Sigma_3$-reflection
- Every uplifting cardinal is a limit of $\Sigma_3$-reflecting cardinals, and is itself $\Sigma_3$-reflecting.
- If $\kappa$ is the least uplifting cardinal, then $\kappa$ is not $\Sigma_4$-reflecting, and there are no $\Sigma_4$-reflecting cardinals below $\kappa$.
The analogous observation for pseudo uplifting cardinals holds as well, namely, every pseudo uplifting cardinal is $\Sigma_3$-reflecting and a limit of $\Sigma_3$-reflecting cardinals; and if $\kappa$ is the least pseudo uplifting cardinal, then $\kappa$ is not $\Sigma_4$-reflecting, and there are no $\Sigma_4$-reflecting cardinals below $\kappa$.
Uplifting Laver functions
Every uplifting cardinal admits an ordinal-anticipating Laver function, and indeed, a HOD-anticipating Laver function, a function $\ell:\kappa\to V_\kappa$, definable in $V_\kappa$, such that for any set $x\in\text{HOD}$ and $\theta$, there is an inaccessible cardinal $\gamma$ above $\theta$ such that $V_\kappa\prec V_\gamma$, for which $\ell^*(\kappa)=x$, where $\ell^*$ is the corresponding function defined in $V_\gamma$.
Connection with the resurrection axioms
Many instances of the (weak) resurrection axiom imply that ${\frak c}^V$ is an uplifting cardinal in $L$:
- RA(all) implies that ${\frak c}^V$ is uplifting in $L$.
- RA(ccc) implies that ${\frak c}^V$ is uplifting in $L$.
- wRA(countably closed)+$\neg$CH implies that ${\frak c}^V$ is uplifting in $L$.
- Under $\neg$CH, the weak resurrection axioms for the classes of axiom-A forcing, proper forcing, semi-proper forcing, and posets that preserve stationary subsets of $\omega_1$, respectively, each imply that ${\frak c}^V$ is uplifting in $L$.
Conversely, if $\kappa$ is uplifting, then various resurrection axioms hold in a corresponding lottery-iteration forcing extension.
Theorem. (Hamkins and Johnstone) The following theories are equiconsistent over ZFC:
- There is an uplifting cardinal.
- RA(all)
- RA(ccc)
- RA(semiproper)+$\neg$CH
- RA(proper)+$\neg$CH
- for some countable ordinal $\alpha$, RA($\alpha$-proper)+$\neg$CH
- RA(axiom-A)+$\neg$CH
- wRA(semiproper)+$\neg$CH
- wRA(proper)+$\neg$CH
- for some countable ordinal $\alpha$, wRA($\alpha$-proper})+$\neg$CH
- wRA(axiom-A)+$\neg$CH
- wRA(countably closed)+$\neg$CH
References
- Hamkins, Joel David and Johnstone, Thomas A. Resurrection axioms and uplifting cardinals. www arχiv bibtex