Subtle cardinal

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

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

A cardinal ${\displaystyle \kappa }$ is called ethereal if for every closed and unbounded ${\displaystyle C\subset \kappa }$ and for every sequence ${\displaystyle (A_{\delta })_{\delta <\kappa }}$ of length ${\displaystyle \kappa }$ such that ${\displaystyle A_{\delta }\subset \delta }$ and ${\displaystyle A_{\delta }}$ has the same cardinality as ${\displaystyle \delta }$ for arbitrary ${\displaystyle \delta <\kappa }$, there exist ${\displaystyle \alpha ,\beta }$, belonging to ${\displaystyle C}$, with ${\displaystyle \alpha <\beta }$, such that ${\displaystyle {\textrm {card}}(\alpha )=\mathrm {card} (A_{\beta }\cup A_{\alpha })}$. ^[1]

Subtle cardinals were introduced by Jensen & Kunen (1969). Ethereal cardinals were introduced by Ketonen (1974). Any subtle cardinal is ethereal, ^{[1] p. 388} and any strongly inaccessible ethereal cardinal is subtle. ^{[1] 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 ${\displaystyle \kappa }$ is subtle if and only if in ${\displaystyle V_{\kappa +1}}$, any logic has stationarily many weak compactness cardinals. ^[2]

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 ${\displaystyle \leq \kappa }$ if and only if every transitive set ${\displaystyle S}$ of cardinality ${\displaystyle \kappa }$ contains ${\displaystyle x}$ and ${\displaystyle y}$ such that ${\displaystyle x}$ is a proper subset of ${\displaystyle y}$ and ${\displaystyle x\neq \varnothing }$ and ${\displaystyle x\neq \{\varnothing \}}$. ^{[3] Corollary 2.6} An infinite ordinal ${\displaystyle \kappa }$ is subtle if and only if for every ${\displaystyle \lambda <\kappa }$, every transitive set ${\displaystyle S}$ of cardinality ${\displaystyle \kappa }$ includes a chain (under inclusion) of order type ${\displaystyle \lambda }$.

Extensions

A hypersubtle cardinal is a subtle cardinal which has a stationary set of subtle cardinals below it. ^{[4] p.1014}

See also

• List of large cardinal properties

References

• Friedman, Harvey (2001), "Subtle Cardinals and Linear Orderings", Annals of Pure and Applied Logic, 107 (1–3): 1–34, doi: 10.1016/S0168-0072(00)00019-1
• Jensen, R. B.; Kunen, K. (1969), Some Combinatorial Properties of L and V, Unpublished manuscript

Citations

1. Ketonen, Jussi (1974), "Some combinatorial principles" (PDF), Transactions of the American Mathematical Society , 188, Transactions of the American Mathematical Society, Vol. 188: 387–394, doi: 10.2307/1996785 , ISSN   0002-9947, JSTOR   1996785, MR   0332481
2. W. Boney, S. Dimopoulos, V. Gitman, M. Magidor "Model Theoretic Characterizations of Large Cardinals Revisited"
3. H. Friedman, "Primitive Independence Results" (2002). Accessed 18 April 2024.
4. C. Henrion, "Properties of Subtle Cardinals. Journal of Symbolic Logic, vol. 52, no. 4 (1987), pp.1005--1019."