  Hoare's logic is an axiomatic system of proving programs correct, which has
been extended to be a separation logic to reason about mutable heap structure.
We develop the most fundamental logical structure of strongest postcondition of
Hoare's logic in Peano's arithmetic $PA$. Let $p\in L$ and $S$ be any
while-program. The arithmetical definability of $\textbf{N}$-computable
function $f_S^{\textbf{N}}$ leads to separate $S$ from $SP(p,S)$, which defines
the strongest postcondition of $p$ and $S$ over $\textbf{N}$, achieving an
equivalent but more meaningful form in $PA$. From the reduction of Hoare's
logic to PA, together with well-defined underlying semantics, it follows that
Hoare's logic is sound and complete relative to the theory of $PA$, which is
different from the relative completeness in the sense of Cook. Finally, we
discuss two ways to extend computability from the standard structure to
nonstandard models of $PA$.
