  This paper shows, in the framework of the logical method,the unsolvability of
$k$-set agreement task by devising a suitable formula of epistemic logic. The
unsolvability of $k$-set agreement task is a well-known fact, which is a direct
consequence of Sperner's lemma, a classic result from combinatorial topology.
However, Sperner's lemma does not provide a good intuition for the
unsolvability,hiding it behind the elegance of its combinatorial statement. The
logical method has a merit that it can account for the reason of unsolvability
by a concrete formula, but no epistemic formula for the general unsolvability
result for $k$-set agreement task has been presented so far.
  We employ a variant of epistemic $\mu$-calculus, which extends the standard
epistemic logic with distributed knowledge operators and propositional
fixpoints, as the formal language of logic. With these extensions, we can
provide an epistemic $\mu$-calculus formula that mentions higher-dimensional
connectivity, which is essential in the original proof of Sperner's lemma, and
thereby show that $k$-set agreement tasks are not solvable even by multi-round
protocols. Furthermore, we also show that the same formula applies to establish
the unsolvability for $k$-concurrency, a submodel of the 2-round protocol.
