  For a family $\mathcal{F}$ of sets in $\mathbb{R}^d$, the Krasnoselskii
number of $\mathcal{F}$ is the smallest $m$ such that for any $S \in
\mathcal{F}$, if every $m$ points of $S$ are visible from a common point in
$S$, then any finite subset of $S$ is visible from a single point. More than 35
years ago, Peterson asked whether there exists a Krasnoselskii number for
general sets in $\mathbb{R}^d$. Excluding results for special cases of sets
with strong topological restrictions, the best known result is due to Breen,
who showed that if such a Krasnoselskii number in $\mathbb{R}^2$ exists, then
it is larger than $8$.
  In this paper we answer Peterson's question in the negative by showing that
there is no Krasnoselskii number for the family of all sets in $\mathbb{R}^2$.
The proof is non-constructive, and uses transfinite induction and the well
ordering theorem.
  In addition, we consider Krasnoselskii numbers with respect to visibility
through polygonal paths of length $ \leq n$, for which an analogue of
Krasnoselskii's theorem was proved by Magazanik and Perles. We show, by an
explicit construction, that for any $n \geq 2$, there is no Krasnoselskii
number for the family of general sets in $\mathbb{R}^2$ with respect to
visibility through paths of length $\leq n$. (Here the counterexamples are
finite unions of line segments.)
