# Concerning three classes of Non-Diophantine arithmetics

Michele Caprio, Andrea Aveni and Sayan Mukherjee  Department of Statistical
Science, Duke University, 214 Old Chemistry, Durham, NC 27708-0251
<EMAIL_ADDRESS>https://mc6034.wixsite.com/caprio Department of
Statistical Science, Duke University, 214 Old Chemistry, Durham, NC 27708-0251
<EMAIL_ADDRESS>https://www.researchgate.net/profile/Andrea_Aveni
Department of Statistical Science, Mathematics, Computer Science, and
Biostatistics & Bioinformatics, Duke University, Durham, NC 27708-0251
<EMAIL_ADDRESS>https://sayanmuk.github.io/

###### Abstract.

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.

###### Key words and phrases:

Non-Diophantine arithmetics; convergence of series; paradox of the heap.

###### 2010 Mathematics Subject Classification:

Primary: 03H15; Secondary: 03C62.

##  1\. Introduction.

Although the conventional arithmetic—which we call Diophantine from
Diophantus, the Greek mathematician who first approached this branch of
mathematics—is almost as old as mathematics itself, it sometimes fails to
correctly describe natural phenomena. For example, in [3] Helmoltz points out
that adding one raindrop to another one leaves us with one raindrop, while in
[11] Kline notices that Diophantine arithmetic fails to correctly describe the
result of combining gases or liquids by volume. Indeed, one quarter of alcohol
and one quarter of water only yield about 1.8 quarters of vodka. To overcome
this issue, scholars started developing inconsistent arithmetics, that is,
arithmetics for which one or more Peano axioms were at the same time true and
false. The most striking one was ultraintuitionism, developed by Yesenin-
Volpin in [12], that asserted that only a finite quantity of natural numbers
exists. Other authors suggested that numbers are finite (see e.g. [2] and
[4]), while different scholars adopted a more moderate approach. The
inconsistency of these alternative arithmetics lies in the fact that they are
all grounded in the ordinary Diophantine arithmetic. The first consistent
alternative to Diophantine arithmetic was proposed by Burgin [6], and the name
non-Diophantine seemed perfectly suited for this arithmetic. Non-Diophantine
arithmetics for natural and whole numbers have been studied by Burgin in [6,
7, 8, 9], while those for real and complex numbers by Czachor in [1, 5]. A
complete account on non-Diophantine arithmetics can be found in the recent
book by Burgin and Czachor [10].

There are two types of non-Diophantine arithmetics: dual and projective. In
this paper, we work with the latter. We start by defining an abstract
prearithmetic $\mathbf{A}$,

$\mathbf{A}:=(A,+_{A},\times_{A},\leq_{A}),$

where $A\subset\mathbb{R}$ is the carrier of $\mathbf{A}$ (that is, the set of
the elements of $\mathbf{A}$), $\leq_{A}$ is a partial order on $A$, and
$+_{A}$ and $\times_{A}$ are two binary operations defined on the elements of
$A$. We conventionally call them addition and multiplication, but that can be
any generic operation. Naturally, the conventional Diophantine arithmetic
$\mathbf{R}=(\mathbb{R},+,\times,\leq_{\mathbb{R}})$ of real numbers is an
abstract preartithmetic.111Notice that $+$ is the usual addition, $\times$ is
the usual multiplication, and $\leq_{\mathbb{R}}$ is the usual partial order
on $\mathbb{R}$.

Abstract prearithmetic $\mathbf{A}$ is called weakly projective with respect
to a second abstract prearithmetic $\mathbf{B}=(B,+_{B},\times_{B},\leq_{B})$
if there exist two functions $g:{A}\rightarrow{B}$ and $h:{B}\rightarrow{A}$
such that, for all $a,b\in A$,

$\displaystyle a+_{A}b=h(g(a)+_{B}g(b))\quad\text{ and }\quad
a\times_{A}b=h(g(a)\times_{B}g(b)).$

Function $g$ is called the projector and function $h$ is called the
coprojector for the pair $(\mathbf{A},\mathbf{B})$.

The weak projection of the sum $a+_{B}b$ of two elements of $B$ onto $A$ is
defined as $h(a+_{B}b)$, while the weak projection of the product
$a\times_{B}b$ of two elements of $B$ onto $A$ is defined as
$h(a\times_{B}b)$.

Abstract prearithmetic $\mathbf{A}$ is called projective with respect to
abstract prearithmetic $\mathbf{B}$ if it is weakly projective with respect to
$\mathbf{B}$, with projector $f^{-1}$ and coprojector $f$. We call $f$, that
has to be bijective, the generator of projector and coprojector.

Weakly projective prearithmetics depend on two functional parameters, $g$ and
$h$—one, $f$, if they are projective—and recover the conventional Diophantine
arithmetic when these functions are the identity. To this extent, we can
consider non-Diophantine arithmetics as a generalization of the Diophantine
one.

In this work, we consider three classes of abstract prearithmetics,
$\\{\mathbf{A}_{M}\\}_{M\geq 1}$, $\\{\mathbf{A}_{-M,M}\\}_{M\geq 1}$, and
$\\{\mathbf{B}_{M}\\}_{M>0}$. They are useful to describe some natural and
tech phenomena for which the conventional Diophantine arithmetic fails, and
their elements allow us to overcome the version of the paradox of the heap (or
sorites paradox) stated in [9, Section 2]. The setting of this variant of the
sorites paradox is adding one grain of sand to a heap of sand, and the
question is, once a grain is added, whether the heap is still a heap. The
heart of sorites paradox is the issue of vagueness, in this case vagueness of
the word “heap”.

We show that every element $\mathbf{A}_{M}$ of the first class is a complete
totally ordered semiring, and it is weakly projective with respect to
$\mathbf{R_{+}}$. Furthermore, we prove that the weak projection of any series
$\sum_{n}a_{n}$ of elements of $\mathbb{R}_{+}:=[0,\infty)$ is convergent in
each $\mathbf{A}_{M}$.

The elements $\mathbf{A}_{-M,M}$ of the second class allow to overcome the
paradox of the heap and are weakly projective with respect to the real
Diophantine arithmetic $\mathbf{R}=(\mathbb{R},+,\times,\leq_{\mathbb{R}})$.
The weak projection of any non-oscillating series $\sum_{n}a_{n}$ of terms in
$\mathbb{R}$ is convergent in $\mathbf{A}_{-{M^{\prime}},{M^{\prime}}}$, for
all ${M^{\prime}}\geq 1$. The drawback of working with this class is that its
elements are not semirings, because the addition operation is not associative.

The elements of $\\{\mathbf{B}_{M}\\}_{M>0}$ too can be used to solve the
paradox of the heap. They are complete totally ordered semirings, and are
projective with respect to the extended real Diophantine arithmetic
$\overline{\mathbf{R}}=(\overline{\mathbb{R}},+,\times,\leq_{\overline{\mathbb{R}}})$.
The projection of any non-oscillating series $\sum_{n}a_{n}$ of terms in
$\mathbb{R}$ is convergent in $\mathbf{B}_{{M^{\prime}}}$, for all
${M^{\prime}}>0$.

$\mathbf{A}_{M}$ | $\mathbf{A}_{-M,M}$ | $\mathbf{B}_{M}$
---|---|---
$A_{M}=[0,M]$ | $A_{-M,M}=[-M,M]$ | $B_{M}=[0,M]$
Solves paradox of the heap | Solves paradox of the heap | Solves paradox of the heap
Weakly projective wrt $\mathbf{R}_{+}$ |  | Weakly projective wrt $\mathbf{R}$
---
| Projective wrt $\overline{\mathbf{R}}$
---
Complete totally ordered semiring | Not a semiring | Complete totally ordered semiring
| Weak projection of a series of elements of
---
$\mathbb{R}_{+}$ is convergent
| Weak projection of a series of elements of
---
$\mathbb{R}$ is absolutely convergent
| Projection of a series of elements of
---
$\mathbb{R}$ is absolutely convergent
– |  | Weak projection of a non-oscillating series
---
of elements of $\mathbb{R}$ is convergent
| Projection of a non-oscillating series
---
of elements of $\mathbb{R}$ is convergent

Table 1. A summary of the properties of the classes we introduce in this
paper. All of them can be used to describe those (natural and tech) phenomena
for which Diophantine arithmetics fail.

The paper is divided as follows. Section 2 deals with
$\\{\mathbf{A}_{M}\\}_{M\geq 1}$. Section 3 presents the class
$\\{\mathbf{A}_{-M,M}\\}_{M\geq 1}$, while $\\{\mathbf{B}_{M}\\}_{M>0}$ is
discussed in Section 4.

##  2\. Class $\\{\mathbf{A}_{M}\\}_{M\geq 1}$.

For any real $M\geq 1$, we define the corresponding non-Diophantine
preartithmetic as

$\mathbf{A}_{M}=(A_{M},\oplus,\otimes,\leq_{A_{M}})$

having the following properties:

  * (i)

The order relation $\leq_{A_{M}}$ is the restriction to ${A}_{M}$ of the usual
order on the reals;

  * (ii)

$A_{M}\subset\mathbb{R}_{+}$ has maximal element $M$ and minimal element $0$
with respect to $\leq_{A_{M}}$, and is such that

    * •

$0\in A_{M}$, which ensures having a multiplicative absorbing and additive
neutral element in our set;

    * •

$1\in A_{M}$, which ensures having a multiplicative neutral element in our
set;

    * •

there is at least an element $x\in(0,1)$ such that $x\in A_{M}$;

  * (iii)

It is closed under the following two operations

$\oplus:A_{M}\times A_{M}\rightarrow A_{M},\quad\oplus:(a,b)\mapsto a\oplus
b:=\min(M,a+b)$

and

$\otimes:A_{M}\times A_{M}\rightarrow A_{M},\quad\otimes:(a,b)\mapsto a\otimes
b:=\min(M,a\times b),$

where $+$ and $\times$ denote the usual sum and product in $\mathbb{R}$,
respectively.

######  Proposition 1.

Addition $\oplus$ is associative.

###### Proof.

Pick any $a,b,c\in A_{M}$. Let $a\oplus b=d$, where $d=a+b$ if $a+b\leq M$ or
$d=M$ if $a+b>M$. Let also $b\oplus c=e$, where $e=b+c$ if $b+c\leq M$ or
$e=M$ if $b+c>M$. Then,

$(a\oplus b)\oplus c=d\oplus c=\begin{cases}d+c=(a+b)+c&\text{if }d+c\leq M\\\
M&\text{if }d+c>M\end{cases}$

and

$a\oplus(b\oplus c)=a\oplus e=\begin{cases}a+e=a+(b+c)&\text{if }a+e\leq M\\\
M&\text{if }a+e>M\end{cases}.$

But we know that $(a+b)+c=a+(b+c)$, so the result follows. ∎

Since addition $\oplus$ is associative we have that, for every
$k\in\mathbb{N}$,

$\bigoplus_{n=1}^{k}x_{n}=\min\left(M,\sum_{n=1}^{n}x_{n}\right).$

By imposing on $M$ the relative topology derived from $\mathbb{R}$, we can
define

$\bigoplus^{\infty}_{n=1}x_{n}:=\lim_{k\rightarrow\infty}\bigoplus_{n=1}^{k}x_{n}=\min\left(M,\sum_{n=1}^{\infty}x_{n}\right).$

######  Proposition 2.

$A_{M}=[0,M]$.

###### Proof.

Recall that ${A}_{M}$ is closed under products $\otimes$ and arbitrary
summations $\oplus$. Fix any $x\in(0,1)$ and let $z\in(0,M]$. Then, it is
always possible to express

$z=\bigoplus_{n=1}^{\infty}\left(\bigoplus_{j=1}^{k_{n}}x^{n}\right),$

for some $(k_{n})\in\mathbb{N}^{\mathbb{N}}$, defined recursively as follows

  * •

$k_{1}=\max\\{m\in\mathbb{N}:z>mx\\}$;

  * •

$k_{j+1}=\max\\{m\in\mathbb{N}:z>\sum_{u=1}^{j}k_{u}x^{u}+mx^{j}\\}$.

It is clear that the partial sums are always less than $z\leq M$, and so are
all nonnegative and well defined. Moreover,

$\left|z-\bigoplus_{n=1}^{m}\left(\bigoplus_{j=1}^{k_{n}}x^{n}\right)\right|\leq
x^{m}\rightarrow 0.$

Finally observe that the result of our two operations is always less or equal
than $M$ and, from non-negative numbers, it is impossible to obtain negative
numbers. ∎

This result implies that $\mathbf{A}_{M}$ is a complete totally ordered
semiring. In this Section, we consider the class $\\{\mathbf{A}_{M}\\}_{M\geq
1}$ of abstract prearithmetics.

######  Remark.

Notice that $\mathbf{A}_{M}$ cannot be a ring because for any $a\in
A_{M}\backslash\\{0\\}$, it lacks the additive inverse $-a$; this because we
defined $A_{M}$ to be a subset of $\mathbb{R}_{+}$. Notice also that in this
abstract prearithmetic $M$ is an idempotent element, that is, $M\oplus
M\oplus\cdots\oplus M=M$.

###  2.1. Overcoming the paradox of the heap.

The paradox of the heap is a paradox that arises from vague predicates. A
formulation of such paradox (also called the sorites paradox, from the Greek
word $\sigma\omega\rho o\varsigma$, “heap”), given in [9, Section 2], is the
following.

  * (1)

One million grains of sand make a heap;

  * (2)

If one grain of sand is added to this heap, the heap stays the same;

  * (3)

However, when we add $1$ to any natural number, we always get a new number.

This formulation of the paradox of the heap is proposed by Burgin to inspect
whether adding $$1$ to the assets of a millionaire makes them “more of a
millionaire”, or leaves their fortune unchanged. We use the class
$\\{\mathbf{A}_{M}\\}_{M\geq 1}$ to address paradox of the heap. Indeed, it is
enough to take the element of the class for which $M=1000000$, so that when we
perform the addition $M\oplus 1$, we get $M$. This conveys the idea that
adding a grain of sand to the heap leaves us with a heap.

The abstract prearithmetic we introduced can also be used to describe
phenomena like the one noted by Helmholtz in [3]: adding one raindrop to
another one gives one raindrop, or the one pointed out by Lebesgue (cf. [11]):
putting a lion and a rabbit in a cage, one will not find two animals in the
cage later on. In both these cases, it suffices to consider the element of the
class for which $M=1$, so that $1\oplus 1=1$.

$\mathbf{A}_{M}$ allows us also to avoid introducing inconsistent Diophantine
arithmetics, that is, arithmetics for which one or more Peano axioms were at
the same time true and false. For example, in [2] Rosinger points out that
electronic digital computers, when operating on the integers, act according to
the usual Peano axioms for $\mathbb{N}$ plus an extra ad-hoc axiom, called the
machine infinity axiom. The machine infinity axiom states that there exists
$\breve{M}\in\mathbb{N}$ far greater than $1$ such that
$\breve{M}+1=\breve{M}$ (for example, $\breve{M}=2^{31}-1$ is the maximum
positive value for a $32$-bit signed binary integer in computing). Clearly,
Peano axioms and the machine infinity axiom together give rise to an
inconsistency, which can be easily avoided by working with
$\mathbf{A}_{\breve{M}}$.

In [4], Van Bendegem developed an inconsistent axiomatic arithmetic similar to
the “machine” one described in [2]. He changed the Peano axioms so that a
number that is the successor of itself exists. In particular, the fifth Peano
axiom states that if $x=y$, then $x$ and $y$ are the same number. In the
system of Van Bendegem, starting from some number $n$, all its successors will
be equal to $n$. Then, the statement $n=n+1$ is considered as both true and
false at the same time, giving rise to an inconsistency. It is immediate to
see how this inconsistency can be overcome by working with any abstract
prearithmetic $\mathbf{A}_{M}$ in our class.

###  2.2. $\mathbf{A}_{M}$ is weakly projective with respect to
$\mathbf{R_{+}}$.

Pick any $\mathbf{A}_{M^{\prime}}\in\\{\mathbf{A}_{M}\\}$, and consider
$\mathbf{R_{+}}=(\mathbb{R}_{+},+,\times,\leq_{\mathbb{\mathbb{R}_{+}}})$.
Consider then the functions:

  * •

$g:A_{M}\rightarrow\mathbb{R}_{+}$, $a\mapsto g(a)\equiv a$, so $g$ is the
identity function $\left.\text{Id}\right|_{A_{M}}$;

  * •

$h:\mathbb{R}_{+}\rightarrow A_{M}$, $a\mapsto h(a):=\min(M,a)$.

Now, if we compute $h(g(a)+g(b))$, for all $a,b\in A_{M}$, we have that

$h(g(a)+g(b))=h(a+b)=\min(M,a+b)=a\oplus b.$

Similarly, we show that $h(g(a)\times g(b))=a\otimes b$. Hence, addition and
multiplication in $\mathbb{R}_{+}$ are weakly projected onto addition and
multiplication in $A_{M^{\prime}}$, respectively. So, we can conclude that
$\mathbf{A}_{M^{\prime}}$ is weakly projective with respect to
$\mathbf{R_{+}}$, for all ${M^{\prime}}\geq 1$.

###  2.3. Series of elements of $\mathbf{A}_{M}$.

Consider any series $\bigoplus_{n}a_{n}$ of elements of $A_{M}$. Its
corresponding series $\sum_{n}g(a_{n})$ in $\mathbb{R}_{+}$ can only be
convergent or divergent to $+\infty$. It cannot be divergent to $-\infty$
because we are summing positive elements only, and it cannot be neither
convergent nor divergent (i.e. it cannot oscillate), because the elements of
the series cannot alternate their sign. But since $A_{M}$ has a maximal and a
minimal element, $M$ and $0$ respectively, this means that
$\bigoplus_{n}a_{n}$ is always convergent.

###  2.4. Weak projection of series in $\mathbf{R_{+}}$ onto $\mathbf{A}_{M}$.

In this Section, we show that the weak projection of any series of elements of
$\mathbb{R}_{+}$ converges in $A_{M}$, for all $M\geq 1$. This is an exciting
result because it allows the scholar that needs a particular series to
converge in their analysis to reach that result by performing a weak
projection of the series onto $\mathbf{A}_{M}$, and then continue the analysis
in $\mathbf{A}_{M}$.

Consider any series $\sum_{n}a_{n}$ of elements of $\mathbb{R}_{+}$. Similarly
to what we pointed out before, it can be convergent or divergent to $+\infty$.
It cannot be divergent to $-\infty$ because we are summing positive elements
only, and it cannot be neither convergent nor divergent (i.e. it cannot
oscillate), because the elements of the series cannot alternate their sign.
Let us then show that the weak projection of
$\sum_{n=1}^{\infty}a_{n}:=\lim\limits_{k\rightarrow\infty}\sum_{n=1}^{k}a_{n}$
is convergent.

First, suppose that $\sum_{n=1}^{\infty}a_{n}=L<M$. Then,
$h(\sum_{n=1}^{\infty}a_{n})=h(L)=L$.

Then, let $\sum_{n=1}^{\infty}a_{n}=L\geq M$. We have that
$h(\sum_{n=1}^{\infty}a_{n})=h(L)=M$, where in both cases the last equality
comes from the definition of $h$.

Finally, suppose that $\sum_{n=1}^{\infty}a_{n}=\infty$. Then,

$h\left(\sum_{n=1}^{\infty}a_{n}\right)=h\left(\lim_{k\rightarrow\infty}\sum_{n=1}^{k}a_{n}\right)=\lim_{k\rightarrow\infty}h\left(\sum_{n=1}^{k}a_{n}\right)=M,$

where the second equality comes from $h$ being continuous, and the last
equality comes from the fact that function $h$ is constant once its argument
is equal to $M$. To check that $h$ is continuous, we just need to check that
it is continuous at $M$; but this is immediate, since $\lim_{a\rightarrow
M^{-}}h(a)=M=\lim_{a\rightarrow M^{+}}h(a)$.

The following lemma comes immediately from Proposition 1.

######  Lemma 1.

For any series $\sum_{n=1}^{\infty}a_{n}$ of elements of $\mathbb{R}_{+}$, for
any $k\in\mathbb{N}$,

$h\left(\sum_{n=1}^{\infty}a_{n}\right)=h\left(\sum_{n=1}^{k}a_{n}\right)\oplus
h\left(\sum_{n=k+1}^{\infty}a_{n}\right).$

##  3\. Class $\\{\mathbf{A}_{-M,M}\\}_{M\geq 1}$

For any real $M\geq 1$, we define the corresponding non-Diophantine
prearithmetic as

$\mathbf{A}_{-M,M}=(A_{-M,M},\boxplus,\boxtimes,\leq_{A_{-M,M}})$

having the following properties:

  * (i)

The order relation $\leq_{A_{-M,M}}$ is the restriction to ${A}_{-M,M}$ of the
usual order on the reals;

  * (ii)

$A_{-M,M}\subset\mathbb{R}$ has maximal element $M$ and minimal element $-M$
with respect to $\leq_{A_{-M,M}}$, and is such that

    * •

$0\in A_{-M,M}$, which ensures having a multiplicative absorbing and additive
neutral element in our set;

    * •

$1\in A_{-M,M}$, which ensures having a multiplicative neutral element in our
set;

    * •

$-M<0<M$ and there is at least an element $x\in(0,1)$ such that $x\in
A_{-M,M}$;

    * •

$b\in A_{-M,M}\implies-b\in A_{-M,M}$;

  * (iii)

It is closed under the following two operations

$\displaystyle\boxplus:A_{-M,M}\times A_{-M,M}$ $\displaystyle\rightarrow
A_{-M,M},$ $\displaystyle(a,b)$ $\displaystyle\mapsto a\boxplus
b:=\begin{cases}a+b&\text{if }a+b\in[-M,M]\\\ M&\text{if }a+b>M\\\ -M&\text{if
}a+b<-M\end{cases},$

where $+$ denotes the usual sum in $\mathbb{R}$, and

$\displaystyle\boxtimes:A_{-M,M}\times A_{-M,M}$ $\displaystyle\rightarrow
A_{-M,M},$ $\displaystyle(a,b)$ $\displaystyle\mapsto a\boxtimes
b:=\begin{cases}a\times b&\text{if }a\times b\in[-M,M]\\\ M&\text{if }a\times
b>M\\\ -M&\text{if }a\times b<-M\end{cases},$

where $\times$ denotes the usual product in $\mathbb{R}$.

Since $\mathbf{A}_{-M,M}$ is closed under products $\boxtimes$ and arbitrary
summations $\boxplus$, it follows, along the lines of Proposition 2, that
$A_{-M,M}=[-M,M]$, for all $M\geq 1$. Notice also that from the definition of
addition $\boxplus$ it follows immediately that $\boxplus$ is commutative, but
$\mathbf{A}_{-M,M}=(A_{-M,M},\boxplus,\boxtimes,\leq_{A_{-M,M}})$ is not a
semiring. Indeed, addition $\boxplus$ is not associative. An easy
counterexample is the following: $-2\boxplus(M\boxplus 1)=-2\boxplus M=M-2$,
while $(-2\boxplus M)\boxplus 1=-1\boxplus M=M-1$.

Despite that, the elements of $\\{\mathbf{A}_{-M,M}\\}_{M\geq 1}$ can still be
useful. Any $\mathbf{A}_{-{M^{\prime}},{M^{\prime}}}$ still solves the paradox
of the heap. If we then consider the functions

  * •

$s:A_{-M,M}\rightarrow\mathbb{R}$, $a\mapsto s(a)\equiv a$, so $s$ is the
identity function $\left.\text{Id}\right|_{A_{-M,M}}$;

  * •

$t:\mathbb{R}\rightarrow A_{-M,M}$, $a\mapsto t(a):=\begin{cases}a&\text{if
}a\in[-M,M]\\\ M&\text{if }a>M\\\ -M&\text{if }a<-M\end{cases}$,

it is immediate to see that, for all $M^{\prime}\geq 1$,
$\mathbf{A}_{-{M^{\prime}},{M^{\prime}}}$ is weakly projective with respect to
the real arithmetic $\mathbf{R}=(\mathbb{R},+,\times,\leq_{\mathbb{R}})$, with
projector $s$ and coprojector $t$.

######  Proposition 3.

Pick any real $M\geq 1$, and any sequence $(x_{n})\in\mathbb{R}^{\mathbb{N}}$.
For every $n$, define $y_{n}:=\sum_{n\leq k}x_{n}$. Then, the weak projection
of $(y_{n})$ converges in $\mathbf{A}_{-M,M}$ if and only if one of the
following holds:

  * (i)

$\liminf_{n}y_{n}\geq M$;

  * (ii)

$\lim y_{n}$ exists and belongs to $(-M,M)$;

  * (iii)

$\limsup_{n}y_{n}\leq-M$.

###### Proof.

The fact that $t$ is continuous is obvious. Suppose then that
$\lim_{n}t(y_{n})=t(\lim_{n}y_{n})\in(-M,M)$, where the equality comes from
the continuity of $t$. By definition of $t$, we have that either
$\lim_{n}y_{n}$ belongs to $(-M,M)$, verifying condition (ii), or it does not.
In this latter case, it may be that either $\lim_{n}y_{n}\geq M$, verifying
condition (i), or $\lim_{n}y_{n}\leq-M$, verifying condition (iii).

The fact that condition (ii) implies $\lim_{n}t(y_{n})\in(-M,M)$ is immediate
by the definition of function $t$. Suppose now that
$\liminf_{n}y_{n}=\infty>M$. This implies that $\lim_{n}y_{n}=\infty$. But
then, $\lim_{n}t(y_{n})=t(\lim_{n}y_{n})=M$, where the first equality comes
form the continuity of $t$, and the last equality from the definition of $t$.
A similar argument shows that if $\limsup_{n}y_{n}=-\infty$, then
$\lim_{n}t(y_{n})=-M$. If $\liminf_{n}y_{n}=L\in[M,\infty)$, the previous
argument shows that $\lim_{n}t(y_{n})=M$, while if
$\limsup_{n}y_{n}=-L\in(-\infty,-M]$, the previous argument shows that
$\lim_{n}t(y_{n})=-M$. This concludes the proof. ∎

The following corollary comes immediately from Proposition 3.

######  Corollary 1.

Pick any real $M\geq 1$. The weak projection of any series of elements of
$\mathbb{R}$ is absolutely convergent in $\mathbf{A}_{-M,M}$. In addition, the
weak projection of any series of elements of $\mathbb{R}$ that is either
convergent or divergent (to $+\infty$ or $-\infty$), converges in
$\mathbf{A}_{-M,M}$.

Notice that, because addition $\boxplus$ is not associative, Lemma 1 does not
hold in this class of abstract prearithmetics. This means that there may exist
a series $\sum_{n=1}^{\infty}a_{n}$ of elements of $\mathbb{R}$ such that

$t\left(\sum_{n=1}^{\infty}a_{n}\right)\neq
t\left(\sum_{n=1}^{k}a_{n}\right)\boxplus
t\left(\sum_{n=k+1}^{\infty}a_{n}\right).$

Hence, we need to project the entire series in the exact order we want the
elements to be summed, otherwise Proposition 3 may not hold.

There is a tradeoff in using this class instead of $\\{\mathbf{A}_{M}\\}$. We
still manage to resolve the paradox of the heap, and we further show that the
weak projection of series that diverge also to $-\infty$, converges in
$\mathbf{A}_{-M,M}$, for all $M\geq 1$. The shortcoming, however, is that we
lose associativity, so working with elements of $A_{-M,M}$ may be difficult.
Ultimately, the choice of one or the other will depend on the application the
scholar has in mind.

##  4\. Class $\\{\mathbf{B}_{M}\\}_{M>0}$.

In this Section we present a class of abstract prearithmetics
$\\{\mathbf{B}_{M}\\}_{M>0}$ where every element is a complete totally ordered
semiring, and such that the projection of a convergent or divergent series (to
$+\infty$ or $-\infty$) of elements of $\mathbb{R}$ converges. Its elements
can be used to solve the paradox of the heap.

For every real $M>0$, we define the corresponding non-Diophantine
prearithmetic as

$\mathbf{B}_{M}=(B_{M},\dotplus,\dottimes,\leq_{B_{M}})$

having the following properties:

  * (i)

The order relation $\leq_{B_{M}}$ is the restriction to $B_{M}$ of the usual
order on the reals;

  * (ii)

$B_{M}=[0,M]$;

  * (iii)

Let $\overline{\mathbb{R}}:=[-\infty,\infty]$ and consider the function

$f:\overline{\mathbb{R}}\rightarrow B_{M},\quad x\mapsto
f(x):=\begin{cases}M\left(\frac{\arctan(x)}{\pi}+\frac{1}{2}\right)&\text{if
}x\in\mathbb{R}\\\ M&\text{if }x=\infty\\\ 0&\text{if }x=-\infty\end{cases}$

and its inverse

$f^{-1}:B_{M}\rightarrow\overline{\mathbb{R}},\quad x\mapsto
f^{-1}(x):=\begin{cases}\tan\left(\frac{\pi}{M}\left(x-\frac{M}{2}\right)\right)&\text{if
}x\in(0,M)\\\ \infty&\text{if }x=M\\\ -\infty&\text{if }x=0\end{cases}.$

Then, $\mathbf{B}_{M}$ is closed under the following two operations

$\dotplus:B_{M}\times B_{M}\rightarrow B_{M},\quad(a,b)\mapsto a\dotplus
b:=f\left(f^{-1}(a)+f^{-1}(b)\right),$

where $+$ denotes the sum in $\overline{\mathbb{R}}$, and

$\dottimes:B_{M}\times B_{M}\rightarrow B_{M},\quad(a,b)\mapsto a\dottimes
b:=f\left(f^{-1}(a)\times f^{-1}(b)\right),$

where $\times$ denotes the product in $\overline{\mathbb{R}}$.

Notice that we do not “force” $0$ and $M$ to be the boundary elements of
$B_{M}$; they come naturally from the way addition $\dotplus$ and
multiplication $\dottimes$ are defined. In addition, we have that by
construction $\mathbf{B}_{M}$ is projective with respect to
$\overline{\mathbf{R}}=(\overline{\mathbb{R}},+,\times,\leq_{\overline{\mathbb{R}}})$,
and that its generator induces an homeomorphism between
$\overline{\mathbb{R}}$ and $[0,M]$. This tells us immediately that
$(B_{M},\dotplus,\dottimes,\leq_{B_{M}})$ is a complete totally ordered
semiring, so addition $\dotplus$ and multiplication $\dottimes$ are
associative. The fact that $+\infty$ and $-\infty$ in $\overline{\mathbb{R}}$
correspond to $M$ and $0$, respectively, tells us that the projection
$f(\sum_{n}a_{n})$ of any series $\sum_{n}a_{n}$ of elements of $\mathbb{R}$
converges in $\mathbf{B}_{M}$, as long as $\sum_{n}a_{n}$ does not oscillate.
The elements of $\mathbf{B}_{M}$ can be used to solve the paradox of the heap;
to see this, notice that $M\dotplus a=M$, for all $a\in B_{M}$ and all $M>0$.
Also, $M$ is an idempotent element of $B_{M}$, for all $M>0$: $M\dotplus
M\dotplus\cdots\dotplus M=M$.

## References

  * [1] Diederik Aerts, Marek Czachor, and Maciej Kuna.  Fourier Transforms on Cantor Sets: A Study in Non-Diophantine Arithmetic and Calculus.  Chaos, Solitons & Fractals, 91:461–468, 2016.
  * [2] Elemer E. Rosinger.  On the Safe Use of Inconsistent Mathematics.  Available at arXiv:0811.2405, 2008.
  * [3] Hermann von Helmholtz.  Zahlen und Messen in Philosophische Aufsatze.  Fues’s Verlag, Leipzig, pages 17–52, 1887.
  * [4] Jean Paul Van Bendegem.  Strict Finitism as a Viable Alternative in the Foundations of Mathematics.  Logique et Analyse, 37(145):23–40, 1994.
  * [5] Marek Czachor.  Relativity of Arithmetic as a Fundamental Symmetry of Physics.  Quantum Studies: Mathematics and Foundations, 3(2):123–133, 2016\.
  * [6] Mark Burgin.  Non-Classical Models of Natural Numbers.  Russian Mathematical Surveys, 32:209–210, 1977.
  * [7] Mark Burgin.  Elements of Non-Diophantine Arithmetics.  Proceedings of the 6th Annual International Conference on Statistics, Mathematics and Related Fields, pages 190–203, 2007.
  * [8] Mark Burgin.  On Weak Projectivity in Arithmetic.  European Journal of Pure and Applied Mathematics, 12(4):1787–1810, 2019.
  * [9] Mark Burgin and Gunter Meissner.  1 + 1 = 3: Synergy Arithmetic in Economics.  Applied Mathematics, 08(02):133–144, 2017.
  * [10] Mark Burgin and Marek Czachor.  Non-Diophantine Arithmetics in Mathematics, Physics and Psychology.  World Scientific, Singapore, 2020.
  * [11] Morris Kline.  Mathematics: The Loss of Certainty.  Oxford University Press, New York, 1980.
  * [12] Alexander C. Yesenin-Volpin.  On the Grounding of Set Theory.  In In: Application of Logic in Science and Technology (in Russian), pages 22–118. Moscow, 1960.