Difference between revisions of "Worldly"

From Cantor's Attic
Jump to: navigation, search
m (Added fact of consistency strength)
Line 11: Line 11:
  
 
A cardinal $\kappa$ is ''$1$-worldly'' if it is worldly and a limit of worldly cardinals. More generally, $\kappa$ is ''$\alpha$-worldly'' if it is worldly and for every $\beta\lt\alpha$, the $\beta$-worldly cardinals are unbounded in $\kappa$.  The cardinal $\kappa$ is ''hyper-worldly'' if it is $\kappa$-worldly. One may proceed to define notions of $\alpha$-hyper-worldly and $\alpha$-hyper${}^\beta$-worldly in analogy with the [[inaccessible#hyper-inaccessible | hyper-inaccessible cardinals]]. Every [[inaccessible]] cardinal $\kappa$ is hyper${}^\kappa$-worldly, and a limit of such kinds of cardinals.
 
A cardinal $\kappa$ is ''$1$-worldly'' if it is worldly and a limit of worldly cardinals. More generally, $\kappa$ is ''$\alpha$-worldly'' if it is worldly and for every $\beta\lt\alpha$, the $\beta$-worldly cardinals are unbounded in $\kappa$.  The cardinal $\kappa$ is ''hyper-worldly'' if it is $\kappa$-worldly. One may proceed to define notions of $\alpha$-hyper-worldly and $\alpha$-hyper${}^\beta$-worldly in analogy with the [[inaccessible#hyper-inaccessible | hyper-inaccessible cardinals]]. Every [[inaccessible]] cardinal $\kappa$ is hyper${}^\kappa$-worldly, and a limit of such kinds of cardinals.
 +
 +
The consistency strength of a $1$-worldly cardinal is stronger than that of a worldly cardinal, the consistency strength of a $2$-worldly cardinal is stronger than that of a $1$-worldly cardinal, etc.
  
 
The worldly cardinal terminology was introduced in lectures of J. D. Hamkins at the CUNY Graduate Center and at NYU.
 
The worldly cardinal terminology was introduced in lectures of J. D. Hamkins at the CUNY Graduate Center and at NYU.

Revision as of 13:28, 14 December 2017

A cardinal $\kappa$ is worldly if $V_\kappa$ is a model of $\text{ZFC}$. It follows that $\kappa$ is a strong limit, a beth fixed point and a fixed point of the enumeration of these, and more.

  • Every inaccessible cardinal is worldly.
  • Nevertheless, the least worldly cardinal is singular and hence not inaccessible.
  • The least worldly cardinal has cofinality $\omega$.
  • Indeed, the next worldly cardinal above any ordinal, if any exist, has cofinality $\omega$.
  • Any worldly cardinal $\kappa$ of uncountable cofinality is a limit of $\kappa$ many worldly cardinals.

Degrees of worldliness

A cardinal $\kappa$ is $1$-worldly if it is worldly and a limit of worldly cardinals. More generally, $\kappa$ is $\alpha$-worldly if it is worldly and for every $\beta\lt\alpha$, the $\beta$-worldly cardinals are unbounded in $\kappa$. The cardinal $\kappa$ is hyper-worldly if it is $\kappa$-worldly. One may proceed to define notions of $\alpha$-hyper-worldly and $\alpha$-hyper${}^\beta$-worldly in analogy with the hyper-inaccessible cardinals. Every inaccessible cardinal $\kappa$ is hyper${}^\kappa$-worldly, and a limit of such kinds of cardinals.

The consistency strength of a $1$-worldly cardinal is stronger than that of a worldly cardinal, the consistency strength of a $2$-worldly cardinal is stronger than that of a $1$-worldly cardinal, etc.

The worldly cardinal terminology was introduced in lectures of J. D. Hamkins at the CUNY Graduate Center and at NYU.