Makkai and Paré introduced the following binary relation on regular cardinals: given $\kappa$ and $\lambda$, $\kappa \vartriangleleft \lambda$ (read, $\kappa$ is **sharply less than** $\lambda$) when $\kappa < \lambda$ and, for every set $X$ of cardinality $< \lambda$, the set $P\_\kappa (X)$ of all subsets of $X$ of cardinality $< \kappa$ has a cofinal subset of cardinality $< \lambda$.


It is not hard to see that $\kappa \vartriangleleft \kappa^+$ for all regular cardinals $\kappa$. On the other hand, $\aleph\_1$ is *not* sharply less than $\aleph\_{\omega + 1}$, so $\vartriangleleft$ is not the same as $<$. Nonetheless, it is true that $\aleph\_0 \vartriangleleft \lambda$ for every uncountable regular cardinal $\lambda$, simply because $P\_{\aleph\_0} (X)$ has the same cardinality as $X$ when $X$ is infinite. More generally, if for all (not necessarily regular) cardinals $\kappa' < \kappa$ and all cardinals $\lambda' < \lambda$, we have ${\lambda'}^{\kappa'} < \lambda$, then $\kappa \vartriangleleft \lambda$. In particular if $\lambda$ is an inaccessible cardinal then $\kappa \vartriangleleft \lambda$ for all regular cardinals $\kappa < \lambda$.


**Question.** Do there exist *uncountable* regular cardinals $\kappa$ such that $\kappa \vartriangleleft \lambda$ if and only if $\kappa < \lambda$? Is there a proper class of them?



When are all greater cardinals sharply greater?
First note that if $\kappa \vartriangleleft \mu^+$ then $\mu^{\lt\kappa} = |P\_\kappa(\mu)| \leq 2^{\lt\kappa}\cdot\mu$. Therefore $\mu^{\lt\kappa} = \mu$ if $\kappa \vartriangleleft \mu^+$ and $2^{\lt\kappa} \leq \mu$. This is impossible if $\operatorname{cf}(\mu) \lt \kappa$ by [König's Theorem](http://en.wikipedia.org/wiki/K%C3%B6nig%27s_theorem_%28set_theory%29), which implies that $\mu^{\operatorname{cf}(\mu)} \gt \mu$ always holds. Since there are arbitrarily large cardinals $\mu$ with countable cofinality, the only infinite cardinal with this property is $\aleph\_0$.


