Strong cardinals were created as a weakening of [supercompact](Supercompact.md "Supercompact") cardinals introduced by Dodd and Jensen in 1982 {% cite Jech2003 %}. They are defined as a strengthening of [measurability](Measurable.md "Measurable"), being that they are critical points of [elementary embeddings](Elementary%20embedding.md "Elementary embedding") $j:V\rightarrow M$ for some transitive inner model of [ZFC](ZFC.md "ZFC") $M$. [[Hypermeasurable|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. # Definitions of Strongness There are multiple equivalent definitions of strongness, using [[Elementary embedding|elementary embeddings]] and [[extenders]]. ## Elementary Embedding Characterisation 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 $j(\kappa)>\gamma$ and $V_\gamma\subset M$. 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. ### $A$-strongness - 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 $A\cap V_{\kappa+\gamma} = j(A)\cap V_{\kappa+\gamma}$. 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 $\gamma<\delta$. ## Extender Characterisation A cardinal $\kappa$ is **$\gamma$-strong** if there is a $(\kappa,\gamma)$-[[Extenders|extender]] $E$ such that $j_{E}(\kappa)>\gamma$ and $V_{\gamma}\subseteq \mathrm{Ult}(V,E)$. A cardinal $\kappa$ is **strong** iff it is <a href="Uncountable" class="mw-redirect" title="Uncountable">uncountable</a> and for every set $X$ of rank $\lambda\geq\kappa$, there is a $(\kappa,\beth_\lambda^+)$-[[Extenders|extender]] $E$ such that $X\in Ult(V,E)$ and $\lambda\lt j_{E}(\kappa)$. {% cite Jech2003 %} Once again, a more intuitive way to think about strongness is that there are many $(\kappa,\lambda)$-extenders $E$. # Related notions [[Hypermeasurable]] ## $C^{(n)}$-strong Recall that $C^{(n)}$ is the class of $\Sigma_{n}$-[[Reflecting#Reflection and correctness|correct]] cradinals. A cardinal $\kappa$ is $λ$-$C^{(n)}$-strong iff there exists an elementary embedding $j : V → M$ for transitive $M$, with $crit(j) = κ$, $j(κ) \gt λ$, $V_λ ⊆ M$ and $j(κ) ∈ C^{(n)}$. A cardinal $κ$ is $C^{(n)}$-strong iff for every $λ \gt κ$, $κ$ is $λ$-$C^{(n)}$-strong. Equivalently (see {% cite Kanamori2009 %} 26.7), $\kappa$ is $λ$-$C^{(n)}$-strong iff there exists a $(κ, β)$-extender $E$, for some $β \gt \|V_λ\|$, with $V_λ ⊆ M_E$ and $λ \lt j_E(κ) ∈ C^{(n)}$. # Facts about Strongness and Hypermeasurability Here is a list of facts about these cardinals: - A cardinal $\kappa$ is [measurable](Measurable.md "Measurable") if and only if it is $\kappa$-strong (by definition) - 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. - Every [supercompact](Supercompact.md "Supercompact") cardinal $\kappa$ is strong and has $\kappa$ strong cardinals below it, as well as being a stationary limit of $\{\lambda\lt\kappa:\lambda$ is strong$\}$ - The [Mitchell rank and Mitchell order](Mitchell%20rank%20and%20Mitchell%20order.md "Mitchell rank") of any strong cardinal $o(\kappa)=(2^\kappa)^+$. {% cite Jech2003 %} - Any strong cardinal is $\Sigma_2$-[reflecting](Reflecting.md "Reflecting"). {% cite Jech2003 %} - Every strong cardinal is [strongly unfoldable](Unfoldable.md "Unfoldable") and thus [totally indescribable](Indescribable.md "Indescribable"). {% cite Gitman2011 %} Therefore, each of the following is never strong: - The least [measurable](Measurable.md "Measurable") cardinal. - The least $\kappa$ which is $2^\kappa$-[supercompact](Supercompact.md "Supercompact"), the least $\kappa$ which is $2^{2^\kappa}$-[supercompact](Supercompact.md "Supercompact"), etc. - For each $n$, the least $n$-[huge](Huge.md "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 $n\lt\omega$, The least $\kappa$ such that there is some embedding $j:V_{\lambda+n}\prec V_{\kappa+n}$ with critical point $\lambda$ for some $\lambda\lt\kappa$ (see $n$-[extendible](Extendible.md "Extendible")). - The least $\kappa$ which is both $2^\kappa$-supercompact and [Vopěnka](Vopenka.md "Vopenka"), 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 $\lambda\lt\kappa$, 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](Superstrong.md "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. - Every $λ$-strong cardinal is $λ$-$C^{(n)}$-strong for all $n$. Hence, every strong cardinal is $C^{(n)}$-strong for all $n$. {% cite Bagaria2012 %} # Core Model up to Strongness Dodd and Jensen created a [[Core_model|core model]] based on sequences of <a href="index.php?title=Extender&amp;action=edit&amp;redlink=1" class="new" title="Extender (page does not exist)">extenders</a> of strong cardinals. They constructed a sequence of extenders $\mathcal{E}$ such that: {% cite Jech2003 %} - $L[\mathcal{E}]$ is an inner model of [ZFC](ZFC.md "ZFC"). - $L[\mathcal{E}]$ satisfies <a href="GCH" class="mw-redirect" title="GCH">GCH</a>, 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 [[Beth]] 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 [[Zero sharp|"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. {% cite Jech2003 %} This real is commonly referred to as *the sharp for a strong cardinal*.