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.
Being strongly uplifting (see further) is boldface variant of being uplifting.
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
Strongly Uplifting
(Information in this section comes from [2])
Strongly uplifting cardinals are precisely strongly pseudo uplifting ordinals, strongly uplifting cardinals with weakly compact targets, superstrongly unfoldable cardinals and almost-hugely unfoldable cardinals.
Definitions
An ordinal is strongly pseudo uplifting iff for every ordinal $θ$ it is strongly $θ$-uplifting, meaning that for every $A⊆V_κ$, there exists some ordinal $λ>θ$ and an $A^*⊆V_λ$ such that $(V_κ;∈,A)≺(V_λ;∈,A^*)$ is a proper elementary extension.
An inaccessible cardinal is strongly uplifting iff for every ordinal $θ$ it is strongly $θ$-uplifting, meaning that for every $A⊆V_κ$, there exists some inaccessible(*) $λ>θ$ and an $A^*⊆V_λ$ such that $(V_κ;∈,A)≺(V_λ;∈,A^*)$ is a proper elementary extension. By replacing starred "inaccessible" with "weakly compact" and other properties, we get strongly uplifting with weakly compact etc.
A cardinal $\kappa$ is $\theta$-superstrongly unfoldable iff for every $A\subseteq\kappa$, there is some transitive $M$ with $A\in M\models\text{ZFC}$ and some $j:M\rightarrow N$ an elementary embedding with critical point $\kappa$ such that $j(\kappa)\geq\theta$ and $V_{j(\kappa)}\subseteq N$.
A cardinal $\kappa$ is $\theta$-almost-hugely unfoldable iff for every $A\subseteq\kappa$, there is some transitive $M$ with $A\in M\models\text{ZFC}$ and some $j:M\rightarrow N$ an elementary embedding with critical point $\kappa$ such that $j(\kappa)\geq\theta$ and $N^{<j(\kappa)}\subseteq N$.
$κ$ is then called superstrongly unfoldable (resp. almost-hugely unfoldable) iff it is $θ$-strongly unfoldable (resp. $θ$-almost-hugely unfoldable) for every $θ$; i.e. the target of the embedding can be made arbitrarily large.
Equivalence
For any ordinals $κ$, $θ$, the following are equivalent:
- $κ$ is strongly pseudo $(θ+1)$-uplifting.
- $κ$ is strongly $(θ+1)$-uplifting.
- $κ$ is strongly $(θ+1)$-uplifting with weakly compact targets.
- $κ$ is strongly $(θ+1)$-uplifting with totally indescribable targets, and indeed with targets having any property of $κ$ that is absolute to all models $V_γ$ with $γ > κ, θ$.
For any cardinal $κ$ and ordinal $θ$, the following are equivalent:
- $κ$ is strongly $(θ+1)$-uplifting.
- $κ$ is superstrongly $(θ+1)$-unfoldable.
- $κ$ is almost-hugely $(θ+1)$-unfoldable.
- For every set $A ∈ H_{κ^+}$ there is a $κ$-model $M⊨\mathrm{ZFC}$ with $A∈M$ and $V_κ≺M$ and a transitive set $N$ with an elementary embedding $j:M→N$ having critical point $κ$ with $j(κ)> θ$ and $V_{j(κ)}≺N$, such that $N^{<j(κ)}⊆N$ and $j(κ)$ is inaccessible, weakly compact and more in $V$.
- $κ^{<κ}=κ$ holds, and for every $κ$-model $M$ there is an elementary embedding $j:M→N$ having critical point $κ$ with $j(κ)> θ$ and $V_{j(κ)}⊆N$, such that $N^{<j(κ)}⊆N$ and $j(κ)$ is inaccessible, weakly compact and more in $V$.
Relations to other cardinals
- If $δ$ is a subtle cardinal, then the set of cardinals $κ$ below $δ$ that are strongly uplifting in $V_δ$ is stationary.
- If $0^♯$ exists, then every Silver indiscernible is strongly uplifting in $L$.
- In $L$, $κ$ is strongly uplifting iff it is unfoldable with cardinal targets.
- Every strongly uplifting cardinal is strongly uplifting in $L$. Every strongly $θ$-uplifting cardinal is strongly $θ$-uplifting in $L$.
- Every strongly uplifting cardinal is strongly unfoldable of every ordinal degree $α$ and a stationary limit of cardinals that are strongly unfoldable of every ordinal degree and so on.
Relation to boldface resurrection axiom
The following theories are equiconsistent over $\mathrm{ZFC}$:
- There is a strongly uplifting cardinal.
- The boldface resurrection axiom for all forcing, for proper forcing, for semi-proper forcing and for c.c.c. forcing.
- The weak boldface resurrection axioms for countably-closed forcing, for axiom-$A$ forcing, for proper forcing and for semi-properforcing, respectively, plus $¬\mathrm{CH}$.
Weakly superstrong cardinal
(Information in this section comes from [3])
Hamkins and Johnstone called an inaccessible cardinal $κ$ weakly superstrong if for every transitive set $M$ of size $κ$ with $κ∈M$ and $M^{<κ}⊆M$, a transitive set $N$ and an elementary embedding $j:M→N$ with critical point $κ$, for which $V_{j(κ)}⊆N$, exist.
It is called weakly almost huge if for every such $M$ there is such $j:M→N$ for which $N^{<j(κ)}⊆N$.
(As usual one can call $j(κ)$ the target.)
A cardinal is superstrongly unfoldable if it is weakly superstrong with arbitrarily large targets, and it is almost hugely unfoldable if it is weakly almost huge with arbitrarily large targets.
If $κ$ is weakly superstrong, it is $0$-extendible and $\Sigma_3$-extendible. Weakly almost huge cardinals also are $\Sigma_3$-extendible. Because $\Sigma_3$-extendibility always can be destroed, all these cardinal properties (among others) are never Lever indestructible.
References
- Hamkins, Joel David and Johnstone, Thomas A. Resurrection axioms and uplifting cardinals. , 2014. www arχiv bibtex
- Hamkins, Joel David and Johnstone, Thomas A. Strongly uplifting cardinals and the boldface resurrection axioms. , 2014. www arχiv bibtex
- Bagaria, Joan and Hamkins, Joel David and Tsaprounis, Konstantinos and Usuba, Toshimichi. Superstrong and other large cardinals are never {Laver} indestructible. Archive for Mathematical Logic 55(1-2):19--35, 2013. www arχiv bibtex