  We study reflection principles of Peano Arithmetic PA which are based on both
proof and provability. Any such reflection principle in PA is equivalent to
either $\Box P\!\rightarrow\! P$ ($\Box P$ stands for `$P$ is provable') or
$\Box^k u\!\!:\!\!P\!\rightarrow\! P$ for some $k\geq 0$ ($t:P$ states `$t$ is
a proof of $P$'). Reflection principles constitute a non-collapsing hierarchy
with respect to their deductive strength $$u\!\!:\!\!P\!\rightarrow\! P\ \
\prec\ \ \Box u\!\!:\!\!P\!\rightarrow\! P\ \ \prec\ \ \Box^2
u\!\!:\!\!P\!\rightarrow\! P \ \ \prec\ \ldots\ \prec\ \ \Box P\!\rightarrow\!
P.$$
