Strong cardinal

• Definitions of Strongness
  • Elementary Embedding Characterization
  • Extender Characterization
• Definitions of Hypermeasurability
  • Elementary Embedding Characterization
• Facts about Strongness and Hypermeasurability
• Core Model up to Strongness

Strong cardinals were created as a weakening of supercompact cardinals introduced by Dodd and Jensen in 1982 (Jech, 2003). 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 $\lambda<j(\kappa)$. (Jech, 2003)

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 $V\neq L[x]$. (Jech, 2003)
• 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 $o(\kappa)=(2^\kappa)^+$. (Jech, 2003)
• Any strong cardinal is $\Sigma_2$-reflecting. (Jech, 2003)
• Every strong cardinal is strongly unfoldable and thus totally indescribable. (Gitman, 2011) 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 (Kanamori, 2009) 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 $n$.(Bagaria, 2012)

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: (Jech, 2003)

• $L[\mathcal{E}]$ 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. (Jech, 2003) This real is commonly referred to as the sharp for a strong cardinal.

References

1. Jech, T. J. (2003). Set Theory (Third). Springer-Verlag. https://logic.wikischolars.columbia.edu/file/view/Jech%2C+T.+J.+%282003%29.+Set+Theory+%28The+3rd+millennium+ed.%29.pdf
2. Gitman, V. (2011). Ramsey-like cardinals. The Journal of Symbolic Logic, 76(2), 519–540. http://boolesrings.org/victoriagitman/files/2011/08/ramseylikecardinals.pdf
3. Kanamori, A. (2009). The higher infinite (Second, p. xxii+536). Springer-Verlag. https://link.springer.com/book/10.1007%2F978-3-540-88867-3
4. Bagaria, J. (2012). \(C^{(n)}\)-cardinals. Archive for Mathematical Logic, 51(3–4), 213–240. https://doi.org/10.1007/s00153-011-0261-8

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