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CONCERNING THREE CLASSES OF NON-DIOPHANTINE

ARITHMETICS

MICHELE CAPRIO, ANDREA AVENI AND SAYAN MUKHERJEE

Abstract. We present three classes of abstract prearithmetics, tAMuMě1,tA´M,M uMě1,

and tBMuMą0. The ﬁrst one is weakly projective with respect to the nonnegative real

Diophantine arithmetic R`“ pR`,`,ˆ,ďR`q, the second one is weakly projective with

respect to the real Diophantine arithmetic R“ pR,`,ˆ,ďRq, while the third one is exactly

projective with respect to the extended real Diophantine arithmetic R“ pR,`,ˆ,ďRq. In

addition, we have that every AMand every BMare a complete totally ordered semiring,

while every A´M,M is not. We show that the projection of any series of elements of R`

converges in AM, for any Mě1, and that the projection of any non-indeterminate series

of elements of Rconverges in A´M,M , for any Mě1, and in BM, for all Mą0. We also

prove that working in AMand in A´M,M , for any Mě1, and in BM, for all Mą0, allows

to overcome a version of the paradox of the heap.

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 [4] Helmoltz points out that adding one raindrop to another one leaves us with one rain-

drop, while in [13] 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 quar-

ter 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 [14], that asserted that only a ﬁnite quantity of natural numbers exists.

Other authors suggested that numbers are ﬁnite (see e.g. [2] and [5]), 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 [7], 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 [7,8,9,10], while those for

real and complex numbers by Czachor in [1,6]. A complete account on non-Diophantine

arithmetics can be found in the recent book by Burgin and Czachor [11]. There, for exam-

ple, the authors show how non-Diophantine arithmetics are crucial for the development and

application of diﬀerent kinds of non-Newtonian calculi [3,6,12].

2010 Mathematics Subject Classiﬁcation. Primary: 03H15; Secondary: 03C62.

Key words and phrases. Non-Diophantine arithmetics; convergence of series; paradox of the heap.

2M. Caprio, A. Aveni and S. Mukherjee

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 A,

A:“ pA, `A,ˆA,ďAq,

where AĂRis the carrier of A(that is, the set of the elements of A), ďAis a partial

order on A, and `Aand ˆAare 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 R“ pR,`,ˆ,ďRqof real numbers is an

abstract preartithmetic.1

Abstract prearithmetic Ais called weakly projective with respect to a second abstract

prearithmetic B“ pB, `B,ˆB,ďBqif there exist two functions g:AÑBand h:BÑA

such that, for all a1, a2PA,

a1`Aa2“hpgpa1q `Bgpa2qq and a1ˆAa2“hpgpa1q ˆBgpa2qq.

Function gis called the projector and function his called the coprojector for the pair pA,Bq.

The weak projection of the sum b1`Bb2of two elements of Bonto Ais deﬁned as hpb1`Bb2q,

while the weak projection of the product b1ˆBb2of two elements of Bonto Ais deﬁned as

hpb1ˆBb2q.

Abstract prearithmetic Ais called exactly projective with respect to abstract prearithmetic

Bif it is weakly projective with respect to B, with projector f´1and coprojector f. We call

f, that has to be bijective, the generator of projector and coprojector.

Weakly projective prearithmetics depend on two functional parameters, gand h– one, f,

if they are exactly 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, tAMuMě1,tA´M,M uMě1,

and tBMuMą0. 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 [10, 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 the same heap. The heart of sorites paradox

is the issue of vagueness, in this case vagueness of the word “heap”.

We show that every element AMof the ﬁrst class is a complete totally ordered semiring, and

it is weakly projective with respect to R`. Furthermore, we prove that the weak projection

of any series řnanof elements of R`:“ r0,8q is convergent in each AM.

The elements 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 R“ pR,`,ˆ,ďRq.

The weak projection of any non-indeterminate series řnanof terms in Ris convergent in

A´M1,M1, for all M1ě1. The drawback of working with this class is that its elements are

not semirings, because the addition operation is not associative.

The elements of tBMuMą0too can be used to solve the paradox of the heap. They are com-

plete totally ordered semirings, and are exactly projective with respect to the extended real

1Notice that `is the usual addition, ˆis the usual multiplication, and ďRis the usual partial order on R.

On three classes of Non-Diophantine arithmetics 3

Diophantine arithmetic R“ pR,`,ˆ,ďRq. The exact projection of any non-indeterminate

series řnanof terms in Ris convergent in BM1, for all M1ą0.

AMA´M,M BM

AM“ r0, M sA´M,M “ r´M, M sBM“ r0, Ms

Solves paradox of the heap Solves paradox of the heap Solves paradox of the heap

Weakly projective wrt R`Weakly projective wrt RExactly projective wrt R

Complete totally ordered semiring Not a semiring Complete totally ordered semiring

Weak projection of a series of elements of

R`is convergent

Weak projection of a series of elements of

Ris absolutely convergent

Exact projection of a series of elements of

Ris absolutely convergent

–Weak projection of a non-indeterminate series

of elements of Ris convergent

Exact projection of a non-indeterminate series

of elements of Ris 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 2deals with tAMuMě1. Section 3presents the

class tA´M,M uMě1, while tBMuMą0is discussed in Section 4.

2. Class tAMuMě1.

For any real Mě1, we deﬁne the corresponding non-Diophantine preartithmetic as

AM“ pAM,‘,b,ďAMq

having the following properties:

(i) The order relation ďAMis the restriction to AMof the usual order on the reals;

(ii) AMĂR`has maximal element Mand minimal element 0with respect to ďAM, and

is such that

•0PAM, which ensures having a multiplicative absorbing and additive neutral

element in our set;

•1PAM, which ensures having a multiplicative neutral element in our set;

•there is at least an element xP p0,1qsuch that xPAM;

(iii) It is closed under the following two operations

‘:AMˆAMÑAM,‘:pa, bq ÞÑ a‘b:“minpM, a `bq

and

b:AMˆAMÑAM,b:pa, bq ÞÑ abb:“minpM, a ˆbq,

where `and ˆdenote the usual sum and product in R, respectively.

Proposition 1. Addition ‘is associative.

Proof. Pick any a, b, c PAM. Let a‘b“d, where d“a`bif a`bďMor d“Mif

a`bąM. Let also b‘c“e, where e“b`cif b`cďMor e“Mif b`cąM. Then,

pa‘bq ‘ c“d‘c“#d`c“ pa`bq ` cif d`cďM

Mif d`cąM

4M. Caprio, A. Aveni and S. Mukherjee

and

a‘ pb‘cq “ a‘e“#a`e“a` pb`cqif a`eďM

Mif a`eąM.

But we know that pa`bq ` c“a` pb`cq, so the result follows.

Since addition ‘is associative we have that, for every kPN,

k

à

n“1

xn“min ˜M,

n

ÿ

n“1

xn¸.

By imposing on Mthe relative topology derived from R, we can deﬁne

8

à

n“1

xn:“lim

kÑ8

k

à

n“1

xn“min ˜M,

8

ÿ

n“1

xn¸.

Proposition 2. AM“ r0, M s.

Proof. Recall that AMis closed under products band arbitrary summations ‘. Fix any

xP p0,1qand let zP p0, Ms. Then, it is always possible to express

z“

8

à

n“1˜kn

à

j“1

xn¸,

for some pknq P NN, deﬁned recursively as follows

‚k1“maxtmPN:ząmxu;

‚kj`1“maxtmPN:ząřj

u“1kuxu`mxju.

It is clear that the partial sums are always less than zďM, and so are all nonnegative and

well deﬁned. Moreover,

ˇˇˇˇˇ

z´

m

à

n“1˜kn

à

j“1

xn¸ˇˇˇˇˇ

ďxmÑ0.

Finally observe that the result of our two operations is always less or equal than Mand,

from non-negative numbers, it is impossible to obtain negative numbers.

This result implies that AMis a complete totally ordered semiring. In this Section, we

consider the class tAMuMě1of abstract prearithmetics.

Remark. Notice that AMcannot be a ring because for any aPAMzt0u, it lacks the additive

inverse ´a; this because we deﬁned AMto be a subset of R`. Notice also that in this abstract

prearithmetic Mis an idempotent element, that is, M‘M‘ ¨ ¨ ¨ ‘ 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 σωρoς , “heap”), given in [10, 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 1to any natural number, we always get a new number.

On three classes of Non-Diophantine arithmetics 5

This formulation of the paradox of the heap is proposed by Burgin to inspect whether adding

$1to the assets of a millionaire makes them “more of a millionaire”, or leaves their fortune

unchanged. We use the class tAMuMě1to 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‘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 [4]: adding one raindrop to another one gives one raindrop, or the

one pointed out by Lebesgue (cf. [13]): 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‘1“1.

AMallows us also to avoid introducing inconsistent Diophantine arithmetics, that is, arith-

metics 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 Nplus an extra ad-hoc axiom, called the machine

inﬁnity axiom. The machine inﬁnity axiom states that there exists ˘

MPNfar greater than 1

such that ˘

M`1“˘

M(for example, ˘

M“231 ´1is 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 A˘

M.

In [5], Van Bendegem developed an inconsistent axiomatic arithmetic similar to the “ma-

chine” 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 xand

yare 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`1is 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 AMin our class.

2.2. AMis weakly projective with respect to R`.Pick any AM1P tAMu, and consider

R`“ pR`,`,ˆ,ďR`q. Consider then the functions:

‚g:AMÑR`,aÞÑ gpaq ” a, so gis the identity function Id|AM;

‚h:R`ÑAM,aÞÑ hpaq:“minpM, aq.

Now, if we compute hpgpaq ` gpbqq, for all a, b PAM, we have that

hpgpaq ` gpbqq “ hpa`bq “ minpM, a `bq “ a‘b.

Similarly, we show that hpgpaq ˆ gpbqq “ abb. Hence, addition and multiplication in R`are

weakly projected onto addition and multiplication in AM1, respectively. So, we can conclude

that AM1is weakly projective with respect to R`, for all M1ě1.

2.3. Series of elements of AM.Consider any series Ànanof elements of AM. Its corre-

sponding series řngpanqin R`can only be convergent or divergent to `8. It cannot be

divergent to ´8 because we are summing positive elements only, and it cannot be neither

convergent nor divergent (i.e. it cannot be indeterminate), because the elements of the series

cannot alternate their sign. But since AMhas a maximal and a minimal element, Mand 0

respectively, this means that Ànanis always convergent.

6M. Caprio, A. Aveni and S. Mukherjee

2.4. Weak projection of series in R`onto AM.In this Section, we show that the weak

projection of any series of elements of R`converges in AM, for all Mě1. 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 AM, and then continue

the analysis in AM.

Consider any series řnanof elements of R`. Similarly to what we pointed out before,

it can be convergent or divergent to `8. It cannot be divergent to ´8 because we are

summing positive elements only, and it cannot be neither convergent nor divergent (i.e. it

cannot be indeterminate), because the elements of the series cannot alternate their sign. Let

us then show that the weak projection of ř8

n“1an:“lim

kÑ8 řk

n“1anis convergent.

First, suppose that ř8

n“1an“LăM. Then, hpř8

n“1anq “ hpLq “ L.

Then, let ř8

n“1an“LěM. We have that hpř8

n“1anq “ hpLq “ M, where in both cases

the last equality comes from the deﬁnition of h.

Finally, suppose that ř8

n“1an“ 8. Then,

h˜8

ÿ

n“1

an¸“h˜lim

kÑ8

k

ÿ

n“1

an¸“lim

kÑ8 h˜k

ÿ

n“1

an¸“M,

where the second equality comes from hbeing continuous, and the last equality comes from

the fact that function his constant once its argument is equal to M. To check that his

continuous, we just need to check that it is continuous at M; but this is immediate, since

limaÑM´hpaq “ M“limaÑM`hpaq.

The following lemma comes immediately from Proposition 1.

Lemma 1. For any series ř8

n“1anof elements of R`, for any kPN,

h˜8

ÿ

n“1

an¸“h˜k

ÿ

n“1

an¸‘h˜8

ÿ

n“k`1

an¸.

3. Class tA´M,M uMě1

For any real Mě1, we deﬁne the corresponding non-Diophantine prearithmetic as

A´M,M “ pA´M ,M ,‘,b,ďA´M,M q

having the following properties:

(i) The order relation ďA´M,M is the restriction to A´M ,M of the usual order on the reals;

(ii) A´M,M ĂRhas maximal element Mand minimal element ´Mwith respect to

ďA´M,M , and is such that

•0PA´M,M , which ensures having a multiplicative absorbing and additive neutral

element in our set;

•1PA´M,M , which ensures having a multiplicative neutral element in our set;

•´Mă0ăMand there is at least an element xP p0,1qsuch that xPA´M,M ;

•bPA´M,M ùñ ´bPA´M,M ;

On three classes of Non-Diophantine arithmetics 7

(iii) It is closed under the following two operations

‘:A´M,M ˆA´M,M ÑA´M,M ,

pa, bq ÞÑ a‘b:“$

’

&

’

%

a`bif a`bP r´M, M s

Mif a`bąM

´Mif a`bă ´M

,

where `denotes the usual sum in R, and

b:A´M,M ˆA´M,M ÑA´M,M ,

pa, bq ÞÑ abb:“$

’

&

’

%

aˆbif aˆbP r´M, M s

Mif aˆbąM

´Mif aˆbă ´M

,

where ˆdenotes the usual product in R.

Since A´M,M is closed under products band arbitrary summations ‘, it follows, along

the lines of Proposition 2, that A´M,M “ r´M, M s, for all Mě1. Notice also that from

the deﬁnition of addition ‘it follows immediately that ‘is commutative, but A´M,M “

pA´M,M ,‘,b,ďA´M,M qis not a semiring. Indeed, addition ‘is not associative. An easy

counterexample is the following: ´2‘pM‘1q “ ´2‘M“M´2, while p´2‘Mq‘1“

´1‘M“M´1.

Despite that, the elements of tA´M,M uMě1can still be useful. Any A´M1,M1still solves

the paradox of the heap. If we then consider the functions

‚s:A´M,M ÑR,aÞÑ spaq ” a, so sis the identity function Id|A´M,M ;

‚t:RÑA´M,M ,aÞÑ tpaq:“$

’

&

’

%

aif aP r´M, M s

Mif aąM

´Mif aă ´M

,

it is immediate to see that, for all M1ě1,A´M1,M1is weakly projective with respect to the

real arithmetic R“ pR,`,ˆ,ďRq, with projector sand coprojector t.

Proposition 3. Pick any real Mě1, and any sequence pxnq P RN. For every n, deﬁne

yn:“řnďkxn. Then, the weak projection of pynqconverges in A´M,M if and only if one of

the following holds:

(i) lim infnyněM;

(ii) lim ynexists and belongs to p´M, Mq;

(iii) lim supnynď ´M.

Proof. The fact that tis continuous is obvious. Suppose then that limntpynq “ tplimnynq P

p´M, M q, where the equality comes from the continuity of t. By deﬁnition of t, we have

that either limnynbelongs to p´M, Mq, verifying condition (ii), or it does not. In this latter

case, it may be that either limnyněM, verifying condition (i), or limnynď ´M, verifying

condition (iii).

The fact that condition (ii) implies limntpynq P p´M, M qis immediate by the deﬁnition of

function t. Suppose now that lim infnyn“8ąM. This implies that limnyn“ 8. But then,

8M. Caprio, A. Aveni and S. Mukherjee

limntpynq “ tplimnynq “ 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 lim supnyn“ ´8,

then limntpynq “ ´M. If lim infnyn“LP rM, 8q, the previous argument shows that

limntpynq “ M, while if lim supnyn“ ´LP p´8,´Ms, the previous argument shows that

limntpynq “ ´M. This concludes the proof.

The following corollary comes immediately from Proposition 3.

Corollary 1. Pick any real Mě1. The weak projection of any series of elements of Ris

absolutely convergent in A´M,M . In addition, the weak projection of any series of elements

of Rthat is either convergent or divergent (to `8 or ´8), converges in A´M,M .

Notice that, because addition ‘is not associative, Lemma 1does not hold in this class of

abstract prearithmetics. This means that there may exist a series ř8

n“1anof elements of R

such that

t˜8

ÿ

n“1

an¸‰t˜k

ÿ

n“1

an¸‘t˜8

ÿ

n“k`1

an¸.

Hence, we need to project the entire series in the exact order we want the elements to be

summed, otherwise Proposition 3may not hold.

There is a tradeoﬀ in using this class instead of tAMu. We still manage to resolve the

paradox of the heap, and we further show that the weak projection of series that diverge

also to ´8, converges in A´M,M , for all Mě1. 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 tBMuMą0.

In this Section we present a class of abstract prearithmetics tBMuMą0where every element

is a complete totally ordered semiring, and such that the projection of a convergent or

divergent series (to `8 or ´8) of elements of Rconverges. Its elements can be used to solve

the paradox of the heap.

For every real Mą0, we deﬁne the corresponding non-Diophantine prearithmetic as

BM“ pBM,`,ˆ,ďBMq

having the following properties:

(i) The order relation ďBMis the restriction to BMof the usual order on the reals;

(ii) BM“ r0, Ms;

(iii) Let R:“ r´8,8s and consider the function

f:RÑBM, x ÞÑ fpxq:“$

’

&

’

%

M´arctanpxq

π`1

2¯if xPR

Mif x“ 8

0if x“ ´8

On three classes of Non-Diophantine arithmetics 9

and its inverse

f´1:BMÑR, x ÞÑ f´1pxq:“$

’

&

’

%

tan `π

M`x´M

2˘˘ if xP p0, Mq

8if x“M

´8 if x“0

.

Then, BMis closed under the following two operations

`:BMˆBMÑBM,pa, bq ÞÑ a`b:“f`f´1paq ` f´1pbq˘,

where `denotes the sum in R, and

ˆ:BMˆBMÑBM,pa, bq ÞÑ aˆb:“f`f´1paq ˆ f´1pbq˘,

where ˆdenotes the product in R.

Notice that we do not “force” 0and Mto be the boundary elements of BM; they come

naturally from the way addition `and multiplication ˆare deﬁned. In addition, we have

that by construction BMis exactly projective with respect to R“ pR,`,ˆ,ďRq, and that its

generator induces an homeomorphism between Rand r0, Ms. This tells us immediately that

pBM,`,ˆ,ďBMqis a complete totally ordered semiring, so addition `and multiplication ˆ

are associative. The fact that `8 and ´8 in Rcorrespond to Mand 0, respectively, tells

us that the exact projection fpřnanqof any series řnanof elements of Rconverges in BM,

as long as řnanis not indeterminate. The elements of BMcan be used to solve the paradox

of the heap; to see this, notice that M`a“M, for all aPBMand all Mą0. Also, Mis

an idempotent element of BM, for all Mą0:M`M`¨ ¨ ¨ `M“M.

Acknowledgements

We are particularly grateful to two anonymous referees for their insightful comments and

suggestions.

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] Endre Pap. g-Calculus. Zb. Rad. Prirod.-Mat. Fak. Ser. Mat., 23(1):145–156, 1993.

[4] Hermann von Helmholtz. Zahlen und Messen in Philosophische Aufsatze. Fues’s Verlag, Leipzig, pages

17–52, 1887.

[5] Jean Paul Van Bendegem. Strict Finitism as a Viable Alternative in the Foundations of Mathematics.

Logique et Analyse, 37(145):23–40, 1994.

[6] Marek Czachor. Relativity of Arithmetic as a Fundamental Symmetry of Physics. Quantum Studies:

Mathematics and Foundations, 3(2):123–133, 2016.

[7] Mark Burgin. Non-Classical Models of Natural Numbers. Russian Mathematical Surveys, 32:209–210,

1977.

[8] Mark Burgin. Elements of Non-Diophantine Arithmetics. Proceedings of the 6th Annual International

Conference on Statistics, Mathematics and Related Fields, pages 190–203, 2007.

[9] Mark Burgin. On Weak Projectivity in Arithmetic. European Journal of Pure and Applied Mathematics,

12(4):1787–1810, 2019.

[10] Mark Burgin and Gunter Meissner. 1 + 1 = 3: Synergy Arithmetic in Economics. Applied Mathematics,

08(02):133–144, 2017.

10 M. Caprio, A. Aveni and S. Mukherjee

[11] Mark Burgin and Marek Czachor. Non-Diophantine Arithmetics in Mathematics, Physics and

Psychology. World Scientiﬁc, Singapore, 2020.

[12] Michael Grossman and Robert Katz. Non-Newtonian calculus. Lee Press, Pigeon Cove, Massachusetts,

1972.

[13] Morris Kline. Mathematics: The Loss of Certainty. Oxford University Press, New York, 1980.

[14] 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.

Department of Statistical Science, Duke University, 214 Old Chemistry, Durham, NC

27708-0251

Email address:michele.caprio@duke.edu

URL:https://mc6034.wixsite.com/caprio

Department of Statistical Science, Duke University, 214 Old Chemistry, Durham, NC

27708-0251

Email address:andrea.aveni@duke.edu

URL: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:sayan@stat.duke.edu

URL:https://sayanmuk.github.io/