Positive $\Sigma$ Operations on Ordinals and Normal Filters on Greatly Mahlo Cardinals
Jech, Thomas
J. Symbolic Logic, Tome 54 (1989) no. 1, p. 226-233 / Harvested from Project Euclid
If $\mathscr{F}$ is a normal filter on a regular uncountable cardinal $\kappa$, let $\|f\|$ be the $\mathscr{F}$-norm of an ordinal function $f$. We introduce the class of positive ordinal operations and prove that if $F$ is a positive operation then $\|F(f)\| \geq F(\|f\|)$. For each $\eta < \kappa^+$ let $f_\eta$ be the canonical $\eta$th function. We show that if $F$ is a $\Sigma$ operation then $F(f_\eta) = f_{F(\eta)}$. As an application we show that if $\kappa$ is greatly Mahlo then there are normal filters on $\kappa$ of order greater than $\kappa^+$.
Publié le : 1989-03-14
Classification: 
@article{1183742863,
     author = {Jech, Thomas},
     title = {Positive $\Sigma$ Operations on Ordinals and Normal Filters on Greatly Mahlo Cardinals},
     journal = {J. Symbolic Logic},
     volume = {54},
     number = {1},
     year = {1989},
     pages = { 226-233},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1183742863}
}
Jech, Thomas. Positive $\Sigma$ Operations on Ordinals and Normal Filters on Greatly Mahlo Cardinals. J. Symbolic Logic, Tome 54 (1989) no. 1, pp.  226-233. http://gdmltest.u-ga.fr/item/1183742863/