Strong cardinal
Strong cardinals were created as a weakening of supercompact cardinals introduced by Dodd and Jensen in 1982 [1]. They are defined as a strengthening of measurability, being that they are critical points of elementary embeddings $j:V\rightarrow M$ for some transitive inner model of ZFC $M$. Hypermeasurability is a weakening of strongness (the property of being a strong cardinal is often called strongness), although if $\lambda=\beth_\lambda$ then a cardinal is $\lambda$-strong iff it is $\lambda$-hypermeasurable.
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Definitions of Strongness
There are multiple equivalent definitions of strongness, using elementary embeddings and extenders.
Elementary Embedding Characterization
A cardinal $\kappa$ is $\gamma$-strong iff it is the critical point of some elementary embedding $j:V\rightarrow M$ for some transitive class $M$ such that $V_\gamma\subset M$. A cardinal $\kappa$ is strong iff it is $\gamma$-strong for each $\gamma$, iff it is $\gamma$-strong for arbitrarily large $\gamma$, iff for each set $x$, $\kappa$ is the critical point of some elementary embedding $j:V\rightarrow M$ for some transitive class $M$ such that $x\in M$.
More intuitively, there are elementary embeddings from $V$ into transitive classes which have critical point $\kappa$ and (in total) contain any set one wishes.
Extender Characterization
A cardinal $\kappa$ is strong iff it is uncountable and for every set $X$ of rank $\lambda\geq\kappa$, there is a $(\kappa,\beth_\lambda^+)$-extender $E$ such that, letting the ultrapower of $V$ by $E$ be called $Ult_E$ and the canonical ultrapower embedding from $V$ to $Ult_E$ be called $j$, $X\in Ult_E$ and $\lambda<j(\kappa)$. [1]
Once again, a more intuitive way to think about strongness is that there are many $(\kappa,\lambda)$-extenders $E$.
Definitions of Hypermeasurability
The definitions of hypermeasurability are very similar to the definitions of strongness, mainly because hypermeasurability is a generalized version of strongness. The intuition behind each definition is also very similar to that of the matching definitions of strongness.
Elementary Embedding Characterization
A cardinal $\kappa$ is $x$-hypermeasurable for a set $x$ iff it is the critical point of some elementary embedding $j:V\rightarrow M$ for some transitive class $M$ such that $x\in M$. A cardinal $\kappa$ is $\lambda$-hypermeasurable iff it is $H_\lambda$-hypermeasurable (where $H_\lambda$ is the set of all sets of hereditary cardinality less than $\lambda$).
Note that a cardinal is $\gamma$-strong iff it is $x$-hypermeasurable for every $x\in V_\gamma$ (iff it is $V_\gamma$-hypermeasurable as well) and a cardinal is strong iff it is $x$-hypermeasurable for every $x$.
Facts about Strongness and Hypermeasurability
Here is a list of facts about these cardinals:
- A cardinal $\kappa$ is $\gamma$-strong if and only if $\kappa$ is $\beth_\gamma$-hypermeasurable, by definition.
- In particular, $\kappa$ is $\mathcal{P}^2(\kappa)$-hypermeasurable if and only if it is $\kappa+2$-strong. This hypothesis appears in many theorems.
- A cardinal $\kappa$ is measurable if and only if it is $\kappa^+$-hypermeasurable, since $\mathcal{P}(\kappa)\subset M$ for any $j:V\to M$ with critical point $\kappa$.
- If there is an $x$-hypermeasurable cardinal, then $V\neq L[x]$. [1]
- Every supercompact cardinal $\kappa$ is strong and has $\kappa$ strong cardinals below it, as well as being a stationary limit of $\{\lambda<\kappa:\lambda$ is strong$\}$
- The Mitchell rank of any strong cardinal $o(\kappa)=(2^\kappa)^+$. [1]
- Any strong cardinal is $\Sigma_2$-reflecting. [1]
- Every Shelah cardinal $κ$ is $ξ$-strong for all $ξ < wt(κ)$ ($ξ$ less than the witnessing number of $κ$).[2]
- Every strong cardinal is strongly unfoldable and thus totally indescribable. [3] Therefore, each of the following is never strong:
- The least measurable cardinal.
- The least $\kappa$ which is $2^\kappa$-supercompact, the least $\kappa$ which is $2^{2^\kappa}$-supercompact, etc.
- For each $n$, the least $n$-huge index cardinal (that is, the least target of an embedding witnessing $n$-hugeness of some cardinal) and the least $n$-huge cardinal.
- For each $n<\omega$, The least $\kappa$ such that there is some embedding $j:V_{\lambda+n}\prec V_{\kappa+n}$ with critical point $\lambda$ for some $\lambda<\kappa$ (see $n$-extendible).
- The least $\kappa$ which is both $2^\kappa$-supercompact and Vopěnka, the least $\kappa$ which is both $2^{2^\kappa}$-supercompact and Vopěnka, etc., the least $\kappa$ which is both measurable and Vopěnka, for each $n$ the least Vopěnka $\kappa$ such that there is some embedding $j:V_{\lambda+n}\prec V_{\kappa+n}$ with critical point $\lambda$ for some $\lambda<\kappa$, and more.
- If there is a strong cardinal then $V\neq L[A]$ for every set $A$.
- Assuming both a strong cardinal and a superstrong cardinal exist, and the least strong cardinal $\kappa$ has a superstrong above it, then the least strong cardinal has $\kappa$ superstrong cardinals below it.
- Every strong cardinal is tall. The existence of a tall cardinal is equiconsistent with the existence of a strong cardinal.
- A cardinal $κ$ is $C^{(n)}$-strong iff for every $λ > κ$, $κ$ is $λ$-$C^{(n)}$-strong, that is, there exists an elementary embedding $j : V → M$ for transitive $M$, with $crit(j) = κ$, $j(κ) > λ$, $V_λ ⊆ M$ and $j(κ) ∈ C^{(n)}$.
- $BMM$ (bounded Martin’s maximum) implies that for every set $X$ there is an inner model with a strong cardinal containing $X$.[6]
- Thus, in particular, $BMM$ implies that for every set $X$, $X^\dagger$ exists.
Core Model up to Strongness
Dodd and Jensen created a core model based on sequences of extenders of strong cardinals. They constructed a sequence of extenders $\mathcal{E}$ such that: [1]
- $L[\mathcal{E}]$ is an inner model of ZFC.
- $L[\mathcal{E}]$ satisfies GCH, the square principle, and the existence of a $\Sigma_3^1$ well-ordering of $\mathbb{R}$.
- $L[\mathcal{E}]$ satisfies that $\mathcal{E}$ witnesses the existence of a strong cardinal
- If there does not exist an inner model of the existence of a strong cardinal, then:
- There is no nontrivial elementary embedding $j:L[\mathcal{E}]\rightarrow L[\mathcal{E}]$
- If $\kappa$ is a singular strong limit cardinal then $(\kappa^+)^{L[\mathcal{E}]}=\kappa^+$
As one can see, $L[\mathcal{E}]$ is a core model up to strongness. Dodd and Jensen also constructed a "sharp" defined analogously to $0^{\#}$, but instead of using $L$ one uses $L[\mathcal{E}]$. They then showed that there is a nontrivial elementary embedding $j:L[\mathcal{E}]\rightarrow L[\mathcal{E}]$ iff such a real exists. [1] This real is commonly referred to as the sharp for a strong cardinal.
$\gamma$-strongness for $A$
Some definitions of Woodin cardinals use the concept of $\gamma$-strongness for $A$: an ordinal $\kappa$ is $\gamma$-strong for $A$ (or $\gamma$-$A$-strong) if there exists a nontrivial elementary embedding $j:V\to M$ with critical point $\kappa$ such that $V_{\kappa+\gamma}\subseteq M$ and $A\cap V_{\kappa+\gamma} = j(A)\cap V_{\kappa+\gamma}$ (also $A\cap H_\gamma = j(A)\cap H_\gamma$[7]). Intuitively, $j$ preserves the part of $A$ that is in $V_{\kappa+\gamma}$. We say that a cardinal $\kappa$ is <$\delta$-$A$-strong if it is $\gamma$-$A$-strong for all $\gamma<\delta$. $\delta$ is Woodin if and only if for every $A\subseteq V_\delta$, there is some $\kappa\lt\delta$ <$\delta$-$A$-strong.
The existence of an embedding that witnesses that $\kappa$ is $\gamma$-$A$-strong is equivalent to the existence of an extender $E ⊂ H_\gamma$ which gives rise to such an embedding through an ultrapower construction.[7]
References
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- Golshani, Mohammad. An Easton like theorem in the presence of Shelah cardinals. M Arch Math Logic 56(3-4):273-287, May, 2017. www DOI bibtex
- Gitman, Victoria. Ramsey-like cardinals. The Journal of Symbolic Logic 76(2):519-540, 2011. www arχiv MR bibtex
- Kanamori, Akihiro. The higher infinite. Second, Springer-Verlag, Berlin, 2009. (Large cardinals in set theory from their beginnings, Paperback reprint of the 2003 edition) www bibtex
- Bagaria, Joan. $C^{(n)}$-cardinals. Archive for Mathematical Logic 51(3--4):213--240, 2012. www arχiv DOI bibtex
- Bagaria, Joan. Axioms of generic absoluteness. Logic Colloquium 2002 , 2006. www DOI bibtex
- Schimmerling, Ernest. Woodin cardinals, Shelah cardinals, and the Mitchell-Steel core model. Proc Amer Math Soc 130(11):3385-3391, 2002. DOI bibtex