cantors-attic

Climb into Cantor’s Attic, where you will find infinities large and small. We aim to provide a comprehensive resource of information about all notions of mathematical infinity.

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Strong cardinal

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:

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)

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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