Easton functions on regular cardinals

An Easton function is a function $F$ such that:

• $\mathrm{dom}(F)$ is a set, or a definable class, of infinite regular cardinals (Cofinality $\mathrm{cf}(\alpha )$, and regular and singular cardinals);
• $F(\kappa )$ is a cardinal for every $\kappa \in \mathrm{dom}(F)$;
• $F$ is nondecreasing: $F(\kappa )\le F(\lambda )$ whenever $\kappa \le \lambda$ are in $\mathrm{dom}(F)$;
• $\mathrm{cf}(F(\kappa ))>\kappa$ for every $\kappa \in \mathrm{dom}(F)$.

Because $\mathrm{cf}(\mu )\le \mu$ for every ordinal $\mu$ ($\mathrm{cf}(\alpha )\le \alpha$; $\mathrm{cf}(0)=0$ and $\mathrm{cf}(\alpha +1)=1$; for a limit ordinal $\lambda$ the value $\mathrm{cf}(\lambda )$ is an infinite cardinal with $\mathrm{cf}(\mathrm{cf}(\lambda ))=\mathrm{cf}(\lambda )$, so it is regular; and every cofinal subset of $\lambda$ has cardinality at least $\mathrm{cf}(\lambda )$, a value that is attained), the last clause forces $\kappa <\mathrm{cf}(F(\kappa ))\le F(\kappa )$, so the values of an Easton function automatically satisfy the strictness $F(\kappa )>\kappa$; the two displayed inequalities together are the classical form (15.7)(i) and (iii) of Necessary constraints on the regular-cardinal continuum function. An Easton function with a set domain is a set-sized Easton function.

The class version is understood over a two-sorted class theory in which the set variables range over the sets of the ground model and $F$ itself is one of the classes; there $F$ is required to be definable from set parameters, its domain is the class of all infinite regular cardinals, and the four clauses above are read with set quantifiers and the class parameter $F$. A proper-class Easton function is not a set of ordered pairs. Its set-sized restrictions specify the cardinal index data for set-sized Easton products; forcing conditions are partial binary-valued functions on the associated triples. The class-theoretic ground assumptions are recorded separately on this page and are not part of the definition of $F$.