  We discuss the finiteness of (two-term) silting objects. First, we
investigate new triangulated categories without silting object. Second, one
studies two classes of $\tau$-tilting-finite algebras and give the numbers of
their two-term silting objects. Finally, we explore when
$\tau$-tilting-finiteness implies representatoin-finiteness, and obtain several
classes of algebras in which a $\tau$-tilting-finite algebra is
representation-finite.
