Difference between revisions of "Strong"

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{{DISPLAYTITLE: Strong cardinal}}
 
{{DISPLAYTITLE: Strong cardinal}}
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[[Category:Large cardinal axioms]]
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[[Category:Critical points]]
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Strong cardinals were created as a weakening of [[supercompact]] cardinals introduced by Dodd and Jensen in 1982 <cite>Jech2003:SetTheory</cite>. They are defined as a strengthening of [[measurable|measurability]], being that they are critical points of [[elementary embedding|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.
  
A cardinal $\kappa$ is ''strong'' if it is $\theta$-strong for every ordinal $\theta$, meaning that there is an elementary embedding $j:V\to M$ of the universe $V$ into a transitive proper class $M$, having critical point $\kappa$ and with $V_\theta\subset M$. One may without loss also suppose that $j(\kappa)\gt\theta$.
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== Definitions of Strongness ==
  
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There are multiple equivalent definitions of strongness, using [[elementary embedding|elementary embeddings]] and [[extender|extenders]].
  
Every strong cardinal is $\Sigma_2$-[[reflecting]], because if any $V_\theta$ satisfies some sentence $\sigma$, then if $j:V\to M$ witnesses $\theta$-strongness, then $M$ satisfies that some ordinal $\theta$ below $j(\kappa)$ has $V_\theta\models\sigma$, and so by elementarity $V$ satisifies that some $\theta\lt\kappa$ has $V_\theta\models\sigma$, as desired.
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=== Elementary Embedding Characterization ===
  
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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 $x\in M$.
  
== Hypermeasurability ==
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More intuitively, there are elementary embeddings from $V$ into transitive classes which have critical point $\kappa$ and (in total) contain any set one wishes.
  
To give a more refined hiearachy, a cardinal $\kappa$ is ''$A$-strong'' or equivalently ''$A$-hypermeasurable'', if there is an elementary embedding $j:V\to M$ of the universe $V$ into a transtive class $M$, with critical point $\kappa$, for which $A\in M$.
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=== Extender Characterization ===
  
When $\delta$ is a cardinal, we say that $\kappa$ is ''$\delta$-hypermeasurable'' to mean really that $\kappa$ is $H_\delta$-strong. (Note, this terminology is technically ambiguous, since one could also interpret $\delta$-hypermeasurability as asserting merely that $\delta\in M$, but this reading is not the intended reading, and the ambiguity is avoided by realizing that $M$ contains all ordinals, and so this latter notion is not needed.)  
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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 $\lambda<j(\kappa)$. <cite>Jech2003:SetTheory</cite>
  
Note that the concept of $\theta$-strong is indexed by ordinals $\theta$ and the concept of $\delta$-hypermeasurable is indexed by cardinals. The latter concept provides a more refined hierarchy, particularly when the GCH fails.  
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Once again, a more intuitive way to think about strongness is that there are many $(\kappa,\lambda)$-extenders $E$.
  
* A cardinal $\kappa$ is $\theta$-strong if and only if $\kappa$ is $\beth_\theta$-hypermeasurable.
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== Definitions of Hypermeasurability ==
* In particular, $\kappa$ is $P^2(\kappa)$-hypermeasurable if and only if it is $\kappa+2$-strong. This hypothesis appears in many theorems.
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* A cardinal $\kappa$ is [[measurable]] if and only if it is $\kappa^+$-hypermeasurable, since $P(\kappa)\subset M$ for any $j:V\to M$ with critical point $\kappa$.
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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.
  
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=== Elementary Embedding Characterization ===
  
== Superstrong cardinal ==
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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|hereditary cardinality]] less than $\lambda$).
  
A cardinal $\kappa$ is ''superstrong'' if there is an elementary embedding $j:V\to M$ of the universe $V$ into a transitive class $M$, with critical point $\kappa$, such that $V_{j(\kappa)}\subset M$.  
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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 $x$.
  
Since the induced [[extender| extenders]] arising from the factors below $j(\kappa)$ are all in $M$, it follows that $M_{j(\kappa)}\models\text{ZFC}+\kappa$ is strong. Thus, the existence of a superstrong cardinal implies the existence of a transitive model of ZFC with a strong cardinal. Therefore, superstrongness strictly exceeds strongness in consistency strength, if the latter is consistent.
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== Facts about Strongness and Hypermeasurability ==
  
Meanwhile, superstrong cardinals need not themselves be strong, and the least superstrong cardinal is definitely not strong. This is simply because superstrongness is witnessed by a single object, the extender of the superstrongness embedding, and thus has complexity $\Sigma_2$, but every strong cardinal is $\Sigma_2$-[[reflecting]]. So if $\kappa$ is strong and hence $\Sigma_2$-reflecting, the existence of a superstrong cardinal must reflect below $\kappa$.
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Here is a list of facts about these cardinals:
  
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*A cardinal $\kappa$ is $\gamma$-strong if and only if $\kappa$ is $\beth_\gamma$-hypermeasurable, by definition.
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*In particular, $\kappa$ is $\mathcal{P}^2(\kappa)$-hypermeasurable if and only if it is $\kappa+2$-strong. This hypothesis appears in many theorems.
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*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 $\kappa$.
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*If there is an $x$-hypermeasurable cardinal, then $V\neq L[x]$. <cite>Jech2003:SetTheory</cite> 
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*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$\}$
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*The [[Mitchell rank]] of any strong cardinal $o(\kappa)=(2^\kappa)^+$. <cite>Jech2003:SetTheory</cite>
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*Any strong cardinal is [[reflecting|$\Sigma_2$-reflecting]]. <cite>Jech2003:SetTheory</cite>
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* Every [[Shelah]] cardinal $κ$ is $ξ$-strong for all $ξ < wt(κ)$ ($ξ$ less than the witnessing number of $κ$).<cite>Golshani2017:EastonLikeInPresenceShelah</cite>
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*Every strong cardinal is [[unfoldable|strongly unfoldable]] and thus [[indescribable|totally indescribable]]. <cite>Gitman2011:RamseyLikeCardinals</cite> Therefore, each of the following is never strong:
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**The least [[measurable]] cardinal.
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**The least $\kappa$ which is [[supercompact|$2^\kappa$-supercompact]], the least $\kappa$ which is [[supercompact|$2^{2^\kappa}$-supercompact]], etc.
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**For each $n$, the least [[huge|$n$-huge]] index cardinal (that is, the least ''target'' of an embedding witnessing $n$-hugeness of some cardinal) and the least $n$-huge cardinal.
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**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 [[extendible|$n$-extendible]]).
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**The least $\kappa$ which is both $2^\kappa$-supercompact and [[Vopenka|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.
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*If there is a strong cardinal then $V\neq L[A]$ for every set $A$.
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*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.
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* Every strong cardinal is [[tall]]. The existence of a tall cardinal is equiconsistent with the existence of a strong cardinal.
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* A cardinal $κ$ is [[correct|$C^{(n)}$-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 $j(κ) ∈ C^{(n)}$.
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** Equivalently (see <cite>Kanamori2009:HigherInfinite</cite> 26.7), $κ$ is $λ$-$C^{(n)}$-strong iff there exists a $(κ, β)$-extender $E$, for some $β > |V_λ|$, with $V_λ ⊆ M_E$ and $λ < j_E(κ) ∈ C^{(n)}$.
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** 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:CnCardinals</cite>
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* [[forcing|$BMM$ (bounded Martin’s maximum)]] implies that for every set $X$ there is an inner model with a strong cardinal containing $X$.<cite>Bagaria2002:AxiomsOfGenericAbsoluteness</cite>
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** Thus, in particular, $BMM$ implies that for every set $X$, [[zero dagger|$X^\dagger$ exists]].
  
{{stub}}
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== Core Model up to Strongness ==
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Dodd and Jensen created a [[core model]] based on sequences of [[extender|extenders]] of strong cardinals. They constructed a sequence of extenders $\mathcal{E}$ such that: <cite>Jech2003:SetTheory</cite>
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*[[L|$L[\mathcal{E}]$]] is an inner model of [[ZFC]].
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*$L[\mathcal{E}]$ satisfies [[GCH]], the square principle, and the existence of a $\Sigma_3^1$ well-ordering of $\mathbb{R}$.
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*$L[\mathcal{E}]$ satisfies that $\mathcal{E}$ witnesses the existence of a strong cardinal
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*If there does not exist an inner model of the existence of a strong cardinal, then:
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**There is no nontrivial elementary embedding $j:L[\mathcal{E}]\rightarrow L[\mathcal{E}]$
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**If $\kappa$ is a singular [[Beth|strong limit]] cardinal then $(\kappa^+)^{L[\mathcal{E}]}=\kappa^+$
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As one can see, $L[\mathcal{E}]$ is a core model up to strongness. Dodd and Jensen also constructed a "sharp" defined analogously to [[Zero sharp|$0^{\#}$]], 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:SetTheory</cite> This real is commonly referred to as ''the sharp for a strong cardinal''.
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== $\gamma$-strongness for $A$ ==
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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 <!--equivalently or not?--> $A\cap H_\gamma = j(A)\cap H_\gamma$<cite>Schimmerling2002:WoodinShelahAndCoreModel</cite>). 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.
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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.<cite>Schimmerling2002:WoodinShelahAndCoreModel</cite>
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{{References}}

Latest revision as of 07:21, 17 November 2019

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.

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 $j(κ) ∈ C^{(n)}$.
    • Equivalently (see [4] 26.7), $κ$ is $λ$-$C^{(n)}$-strong iff there exists a $(κ, β)$-extender $E$, for some $β > |V_λ|$, with $V_λ ⊆ M_E$ and $λ < j_E(κ) ∈ C^{(n)}$.
    • Every $λ$-strong cardinal is $λ$-$C^{(n)}$-strong for all $n$. Hence, every strong cardinal is $C^{(n)}$-strong for all $n$.[5]
  • $BMM$ (bounded Martin’s maximum) implies that for every set $X$ there is an inner model with a strong cardinal containing $X$.[6]

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

  1. Jech, Thomas J. Set Theory. Third, Springer-Verlag, Berlin, 2003. (The third millennium edition, revised and expanded) www   bibtex
  2. 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
  3. Gitman, Victoria. Ramsey-like cardinals. The Journal of Symbolic Logic 76(2):519-540, 2011. www   arχiv   MR   bibtex
  4. 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
  5. Bagaria, Joan. $C^{(n)}$-cardinals. Archive for Mathematical Logic 51(3--4):213--240, 2012. www   arχiv   DOI   bibtex
  6. Bagaria, Joan. Axioms of generic absoluteness. Logic Colloquium 2002 , 2006. www   DOI   bibtex
  7. Schimmerling, Ernest. Woodin cardinals, Shelah cardinals, and the Mitchell-Steel core model. Proc Amer Math Soc 130(11):3385-3391, 2002. DOI   bibtex
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