  In this Part I, we shall prove the consistency of arithmetic without complete
induction from a point of view of strong negation, using its embedding to the
tableau system $\bf SN$ of constructive arithmetic with strong negation without
complete induction, for which two types of cut elimination theorems hold. One
is $\bf SN$-cut elimination theorem for the full system $\bf SN$. The other is
$\bf PCN$-cut elimination theorem for a proposed subsystem $\bf PCN$ of $\bf
SN$. The disjunction property and the E-theorem (existence property) for $\bf
SN$ are also proved. As a novelty, we shall give a simple proof of a restricted
version of $\bf SN$-cut elimination theorem as an application of the
disjunction property, using $\bf PCN$-cut elimination theorem.
