subtle cardinal

In mathematics, subtle cardinals and ethereal cardinals are closely related kinds of large cardinal number.

A cardinal $\kappa$ is called subtle if for every closed and unbounded $C\subset\kappa$ and for every sequence $(A_\delta)_{\delta<\kappa}$ of length $\kappa$ such that $A_\delta\subset\delta$ for all $\delta<\kappa$ (where $A_\delta$ is the $\delta$ th element), there exist $\alpha,\beta$ , belonging to $C$ , with $\alpha<\beta$ , such that $A_\alpha=A_\beta\cap\alpha$ .

A cardinal $\kappa$ is called ethereal if for every closed and unbounded $C\subset\kappa$ and for every sequence $(A_\delta)_{\delta<\kappa}$ of length $\kappa$ such that $A_\delta\subset\delta$ and $A_\delta$ has the same cardinality as $\delta$ for arbitrary $\delta<\kappa$ , there exist $\alpha,\beta$ , belonging to $C$ , with $\alpha<\beta$ , such that $\textrm{card}(\alpha)=\mathrm{card}(A_\beta\cup A_\alpha)$ .{{Citation | last1=Ketonen | first1=Jussi | title=Some combinatorial principles |mr=0332481 | year=1974 | journal=Transactions of the American Mathematical Society | issn=0002-9947 | volume=188 | pages=387–394 | doi=10.2307/1996785 | publisher=Transactions of the American Mathematical Society, Vol. 188 | jstor=1996785| doi-access=free| url=https://www.ams.org/journals/tran/1974-188-00/S0002-9947-1974-0332481-5/S0002-9947-1974-0332481-5.pdf }}

Subtle cardinals were introduced by {{harvtxt|Jensen|Kunen|1969}}. Ethereal cardinals were introduced by {{harvtxt|Ketonen|1974}}. Any subtle cardinal is ethereal,^{p. 388} and any strongly inaccessible ethereal cardinal is subtle.^{p. 391}

Characterizations

Some equivalent properties to subtlety are known.

=Relationship to Vopěnka's Principle=

Subtle cardinals are equivalent to a weak form of Vopěnka cardinals. Namely, an inaccessible cardinal $\kappa$ is subtle if and only if in $V_{\kappa+1}$ , any logic has stationarily many weak compactness cardinals.W. Boney, S. Dimopoulos, V. Gitman, M. Magidor "[https://web.archive.org/web/20231220180708/https://victoriagitman.github.io/files/LargeCardinalLogics.pdf Model Theoretic Characterizations of Large Cardinals Revisited]" (2023).

Vopenka's principle itself may be stated as the existence of a strong compactness cardinal for each logic.

=Chains in transitive sets=

There is a subtle cardinal $\leq\kappa$ if and only if every transitive set $S$ of cardinality $\kappa$ contains $x$ and $y$ such that $x$ is a proper subset of $y$ and $x\neq\varnothing$ and $x\neq\{\varnothing\}$ .H. Friedman, "[https://bpb-us-w2.wpmucdn.com/u.osu.edu/dist/1/1952/files/2014/01/PrimitiveIndResults071302-189vmn0.pdf Primitive Independence Results]" (2002). Accessed 18 April 2024.^{Corollary 2.6} If a cardinal $\lambda$ is subtle, then for every $\alpha<\lambda$ , every transitive set $S$ of cardinality $\lambda$ includes a chain (under inclusion) of order type $\alpha$ .^{Theorem 2.2}

Extensions

A hypersubtle cardinal is a subtle cardinal which has a stationary set of subtle cardinals below it.C. Henrion, "[https://www.jstor.org/stable/2273834 Properties of Subtle Cardinals]. Journal of Symbolic Logic, vol. 52, no. 4 (1987), pp.1005--1019."^p.1014

References

{{citation|first=Harvey |last=Friedman|authorlink=Harvey Friedman (mathematician)|title=Subtle Cardinals and Linear Orderings|journal= Annals of Pure and Applied Logic |year= 2001|volume=107|issue=1–3|pages=1–34

|doi=10.1016/S0168-0072(00)00019-1|doi-access=free}}

{{citation|title=Some Combinatorial Properties of L and V |url=http://www.mathematik.hu-berlin.de/~raesch/org/jensen.html|first=R. B. |last=Jensen|first2=K.|last2=Kunen|publisher=Unpublished manuscript|year=1969}}

=Citations=

Category:Large cardinals