  We present three classes of abstract prearithmetics, $\{\mathbf{A}_M\}_{M
\geq 1}$, $\{\mathbf{A}_{-M,M}\}_{M \geq 1}$, and $\{\mathbf{B}_M\}_{M > 0}$.
The first one is weakly projective with respect to the nonnegative real
Diophantine arithmetic
$\mathbf{R_+}=(\mathbb{R}_+,+,\times,\leq_{\mathbb{R}_+})$, the second one is
weakly projective with respect to the real Diophantine arithmetic
$\mathbf{R}=(\mathbb{R},+,\times,\leq_{\mathbb{R}})$, while the third one is
projective with respect to the extended real Diophantine arithmetic
$\overline{\mathbf{R}}=(\overline{\mathbb{R}},+,\times,\leq_{\overline{\mathbb{R}}})$.
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 projection of any series of elements of $\mathbb{R}_+$ converges
in $\mathbf{A}_M$, for any $M \geq 1$, and that the projection of any
non-oscillating series series of elements of $\mathbb{R}$ converges in
$\mathbf{A}_{-M,M}$, for any $M \geq 1$, and in $\mathbf{B}_M$, for all $M >
0$. We also prove that working in $\mathbf{A}_M$ and in $\mathbf{A}_{-M,M}$,
for any $M \geq 1$, and in $\mathbf{B}_M$, for all $M>0$, allows to overcome a
version of the paradox of the heap.
