  We present three classes of abstract prearithmetics, $\{\mathbf{A}_M\}_{M
\geq 1}$, $\{\mathbf{A}_{-M,M}\}_{M \geq 1}$, and $\{\mathbf{B}_M\}_{M \geq
0}$. The first one is weakly projective with respect to the conventional
nonnegative real Diophantine arithmetic
$\mathbf{R_+}=(\mathbb{R}_+,+,\times,\leq_{\mathbb{R}_+})$, while the other two
are weakly projective with respect to the conventional real Diophantine
arithmetic $\mathbf{R}=(\mathbb{R},+,\times,\leq_{\mathbb{R}})$. In addition,
we have that every $\mathbf{A}_M$ and every $\mathbf{B}_M$ are a complete
totally ordered semiring, while every $\mathbf{A}_{-M,M}$ is not. We show that
the weak projection of any series in $\mathbf{R_+}$ converges in
$\mathbf{A}_M$, for any $M \geq 1$, and that the weak projection of any
non-oscillating series in $\mathbf{R}$ converges in $\mathbf{A}_{-M,M}$, for
any $M \geq 1$, and in $\mathbf{B}_M$, for all $M \in \mathbb{R}_+$. We also
prove that working in $\mathbf{A}_M$ and in $\mathbf{A}_{-M,M}$, for any $M
\geq 1$, allows to overcome a version of the paradox of the heap, while working
in $\mathbf{B}_M$ does not.
