Concerning three classes of non-Diophantine arithmetics
aa r X i v : . [ m a t h . L O ] J a n CONCERNING THREE CLASSES OF NON-DIOPHANTINEARITHMETICS
MICHELE CAPRIO, ANDREA AVENI AND SAYAN MUKHERJEE
Abstract.
We present three classes of abstract prearithmetics, t A M u M ě , t A ´ M,M u M ě ,and t B M u M ě . The first one is weakly projective with respect to the conventional nonneg-ative real Diophantine arithmetic R ` “ p R ` , ` , ˆ , ď R ` q , while the other two are weaklyprojective with respect to the conventional real Diophantine arithmetic R “ p R , ` , ˆ , ď R q .In addition, we have that every A M and every B M are a complete totally ordered semiring,while every A ´ M,M is not. We show that the weak projection of any series in R ` convergesin A M , for any M ě , and that the weak projection of any non-oscillating series in R converges in A ´ M,M , for any M ě , and in B M , for all M P R ` . We also prove thatworking in A M and in A ´ M,M , for any M ě , allows to overcome a version of the paradoxof the heap, while working in B M does not. Introduction.
Although the conventional arithmetic—which we call Diophantine from Diophantus, theGreek mathematician who first approached this branch of mathematics—is almost as old asmathematics 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 [10] Kline notices that Diophantine arithmetic fails to correctly describe the resultof combining gases or liquids by volume. Indeed, one quarter of alcohol and one quarterof water only yield about 1.8 quarters of vodka. To overcome this issue, scholars starteddeveloping inconsistent arithmetics, that is, arithmetics for which one or more Peano axiomswere at the same time true and false. The most striking one was ultraintuitionism, developedby Yesenin-Volpin in [11], 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 scholarsadopted a more moderate approach. The inconsistency of these alternative arithmetics liesin the fact that they are all grounded in the ordinary Diophantine arithmetic. The firstconsistent alternative to Diophantine arithmetic was proposed by Burgin [6], and the namenon-Diophantine seemed perfectly suited for this arithmetic. Non-Diophantine arithmeticsfor natural and whole numbers have been studied by Burgin in [6, 7, 8, 9], while those forreal and complex numbers by Czachor in [1, 5].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 : “ p A, ` A , ˆ A , ď A q , Mathematics Subject Classification.
Primary: 03H15; Secondary: 03C62.
Key words and phrases.
Non-Diophantine arithmetics; convergence of series; paradox of the heap.
M. Caprio, A. Aveni and
S. Mukherjee where A Ă R ` is the carrier of A (that is, the set of the elements of A ), ď A is a partialorder on A , and ` A and ˆ A are two binary operations defined on the elements of A . Weconventionally call them addition and multiplication, but that can be any generic opera-tion. Naturally, the conventional Diophantine arithmetic R ` “ p R ` , ` , ˆ , ď R ` q of positivereal numbers—that coincides with the positive cone of the Banach lattice—is an abstractpreartithmetic, where R ` : “ r , . Abstract prearithmetic A is called weakly projective with respect to a second abstractprearithmetics B “ p B, ` B , ˆ B , ď B q if there exist two functions g : A Ñ B and h : B Ñ A such that, for all a, b P A , a ` A b “ h p g p a q ` B g p b qq and a ˆ A b “ h p g p a q ˆ B g p b qq . Function g is called the projector and function h is called the coprojector for the pair p A , B q .The weak projection of the sum a ` B b of two elements of B onto A is defined as h p a ` B b q ,while the weak projection of the product a ˆ B b of two elements of B onto A is defined as h p a ˆ B b q .Abstract prearithmetic A is called projective with respect to abstract prearithmetic B ifit is weakly projective with respect to B , with projector f ´ 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 thesefunctions are the identity. To this extent, we can consider non-Diophantine arithmetics as ageneralization of the Diophantine one.In this work, we consider three classes of abstract prearithmetics, viz. t A M u M ě , t A ´ M,M u M ě ,and t B M u M P R ` . These classes of prearithmetics are useful to describe some phenomena forwhich the conventional Diophantine arithmetic fails. The former two allow us to overcomethe version of the paradox of the heap (or sorites paradox) stated in [9, Section 2]. Thesetting 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 ofsorites paradox is the issue of vagueness, in this case vagueness of the word heap.We show that every element A M of the class is a complete totally ordered semiring, andit is weakly projective with respect to R ` . Furthermore, we prove that the weak projectionof any series ř n a n of elements of R ` : “ r , is convergent in each A M .The second class, t A ´ M,M u M ě , allows to overcome the paradox of the heap and is such thatevery element A ´ M ,M is weakly projective with respect to the conventional real Diophantinearithmetic R “ p R , ` , ˆ , ď R q . The weak projection of any non-oscillating series ř n a n ofterms in R is convergent in A ´ M ,M , for all M . The drawback of working with this class isthat its elements are not semirings, because the addition operation is not associative.The last one, t B M u M P R ` , is such that every element B M is a semiring and is projectivewith respect to the conventional real Diophantine arithmetic R “ p R , ` , ˆ , ď R q . The weakprojection of any non-oscillating series ř n a n of terms in R is convergent in B M , for all M .The drawback of working with this class is that its elements do not overcome the paradox ofthe heap. Notice that ` is the usual addition, ˆ is the usual multiplication, and ď R ` is the usual partial order on R ` . n three classes of Non-Diophantine arithmetics t A M u M ě . Section 3 presents the class t A ´ M,M u M ě , while t B M u M P R ` is discussed in Section 4. 2. Class t A M u M ě . For any real M ě , we define the corresponding non-Diophantine pre-artithmetic as A M “ p A M , ‘ M , b M , ď M q having the following properties.(i) A M is a subset of R ` ; it contains (the multiplicative absorbing and additive neutralelement), (the multiplicative neutral element), and at least an element x P p , q .In addition, it is closed under the following operations;(ii) ‘ : A M ˆ A M Ñ A M , ‘ : p x, y q ÞÑ min p M, x ` y q ;(iii) b : A M ˆ A M Ñ A M , b : p x, y q ÞÑ min p M, xy q ;(iv) ď M is the restriction to A M of the usual order on the reals. Proposition 1.
Addition ‘ is associative.Proof. Pick any a, b, c P A M . Let a ‘ b “ d , where d “ a ` b if a ` b ď M or d “ M if a ` b ą M . Let also b ‘ c “ e , where e “ b ` c if b ` c ď M or e “ M if b ` c ą M . Then, p a ‘ b q ‘ c “ d ‘ c “ d ` c “ p a ` b q ` c if d ` c ď MM if d ` c ą M and a ‘ p b ‘ c q “ a ‘ e “ a ` e “ a ` p b ` c q if a ` e ď MM if a ` e ą M .
But we know that p a ` b q ` c “ a ` p b ` c q , so the result follows. (cid:3) Since addition ‘ is associative we have that, for every k P N , k à n “ x n “ min ˜ M, n ÿ n “ x n ¸ . By imposing on M the relative topology derived from R , we can define à n “ x n : “ lim k Ñ8 k à n “ x n “ min ˜ M, ÿ n “ x n ¸ . Proposition 2. If A M is closed under products and arbitrary summations, then A M “r , M s .Proof. Fix any x P p , q and let z P p , M s . Then, it is always possible to express z “ à n “ ˜ k n à j “ x n ¸ , for some p k n q P N N , defined recursively as follows ‚ k “ max t m P N : z ą mx u ; M. Caprio, A. Aveni and
S. Mukherjee ‚ k j ` “ max t m P N : z ą ř ju “ k u x u ` mx j u .It is clear that the partial sums are always less than z ď M , and so are all nonnegative andwell defined. Moreover, ˇˇˇˇˇ z ´ m à n “ ˜ k n à j “ x n ¸ˇˇˇˇˇ ď x m Ñ . 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. (cid:3) This result implies that A M is a complete totally ordered semiring. In our work, we aregoing to consider the class t A M u M ě of abstract prearithmetics. Remark.
Notice that A M cannot be a ring because for any a P A M zt u , it lacks the additiveinverse ´ a ; this because we defined A M to be a subset of R ` . Notice also that in this abstractprearithmetic M is 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 thatarises from vague predicates. A formulation of such paradox (also called the sorites paradox,from the Greek word σωρoς , “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 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$ to the assets of a millionaire makes them “more of a millionaire", or leaves their fortuneunchanged. We use the class t A M u M ě to address paradox of the heap. Indeed, it is enoughto take the element of the class for which M “ , so that when we perform the addition M ‘ , we get M . This conveys the idea that adding a grain of sand to the heap leaves uswith a heap.The abstract prearithmetic we introduced can also be used to describe phenomena like theone noted by Helmholtz in [3]: adding one raindrop to another one gives one raindrop, or theone pointed out by Lebesgue (cf. [10]): putting a lion and a rabbit in a cage, one will notfind two animals in the cage later on. In both these cases, it suffices to consider the elementof the class for which M “ , so that ‘ “ . A M allows 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 N plus an extra ad-hoc axiom, called the machineinfinity axiom. The machine infinity axiom states that there exists M P N far greater than such that M ` “ M . Clearly, Peano axioms and the machine infinity axiom together giverise to an inconsistency, which can be easily avoided by working in A M .In [4], 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 thesuccessor 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 , allits successors will be equal to n . Then, the statement n “ n ` is considered as both true n three classes of Non-Diophantine arithmetics A M in our class.2.2. A M is weakly projective with respect to R ` . Pick any A M P t A M u , and consider R ` “ p R ` , ` , ˆ , ď R ` q . Consider then the functions: ‚ g : A M Ñ R ` , a ÞÑ g p a q ” a , so g is the identity function Id | A M ; ‚ h : R ` Ñ A M , a ÞÑ h p a q : “ min p M, a q .Now, if we compute h p g p a q ` g p b qq , for all a, b P A M , we have that h p g p a q ` g p b qq “ h p a ` b q “ min p M, a ` b q “ a ‘ b. Similarly, we show that h p g p a q ˆ g p b qq “ a b b . Hence, addition and multiplication in R ` areweakly projected onto addition and multiplication in A M , respectively. So, we can concludethat A M is weakly projective with respect to R ` , for all M .2.3. Series of elements of A M . Consider any series À n a n of elements of A M . Its corre-sponding series ř n g p a n q in R ` can only be convergent or divergent to `8 . It cannot bedivergent to ´8 because we are summing positive elements only, and it cannot be neitherconvergent nor divergent (i.e. it cannot oscillate), because the elements of the series can-not alternate their sign. But since A M has a maximal and a minimal element, M and respectively, this means that À n a n is always convergent.2.4. Weak projection of series in R ` onto A M . In this subsection, we show that theweak projection of any series of elements of R ` converges in A M , for all M . This is anexciting result because it allows the scholar that needs a particular series to converge in theiranalysis to reach that result by performing a weak projection of the series onto A M , andthen continue the analysis in A M .Consider any series ř n a n of 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 aresumming positive elements only, and it cannot be neither convergent nor divergent (i.e. itcannot oscillate), because the elements of the series cannot alternate their sign. Let us thenshow that the weak projection of ř n “ a n : “ lim k Ñ8 ř kn “ a n is convergent.First, suppose that ř n “ a n “ L ă M . Then, h p ř n “ a n q “ h p L q “ L .Then, let ř n “ a n “ L ě M . We have that h p ř n “ a n q “ h p L q “ M , where in both casesthe last equality comes from the definition of h .Finally, suppose that ř n “ a n “ 8 . Then, h ˜ ÿ n “ a n ¸ “ h ˜ lim k Ñ8 k ÿ n “ a n ¸ “ lim k Ñ8 h ˜ k ÿ n “ a n ¸ “ M, where the second equality comes from h being continuous, and the last equality comes fromthe fact that function h is constant once its argument is equal to M . To check that h iscontinuous, we just need to check that it is continuous at M ; but this is immediate, since lim a Ñ M ´ h p a q “ M “ lim a Ñ M ` h p a q . M. Caprio, A. Aveni and
S. Mukherjee
Lemma 1.
For any series ř n “ a n of elements of R ` , for any k P N , h ˜ ÿ n “ a n ¸ “ h ˜ k ÿ n “ a n ¸ ‘ h ˜ ÿ n “ k ` a n ¸ . Proof.
Immediate from the associativity of addition ‘ . (cid:3) Class t A ´ M,M u M ě . We can define a class t A ´ M,M u M ě of abstract prearithmetics whose elements are definedas: A ´ M,M “ p A ´ M,M , ‘ , b , ď A ´ M,M q , where:(i) The order relation ď A ´ M,M is the restriction to A ´ M,M of the usual order on the reals;(ii) A ´ M,M Ă R has a maximal element M , a minimal element ´ M with respect to ď A ´ M,M , and is such that• P A ´ M,M , which ensures having a multiplicative absorbing and additive neutralelement in our set;• P A ´ M,M , which ensures having a multiplicative neutral element in our set;• ´ M ă ă M and there is at least an element x P p , q such that x P A M ;• b P A M ùñ ´ b P A ´ M,M ;• It is closed under the following operations.(iii) ‘ : A ´ M,M ˆ A ´ M,M Ñ A ´ M,M , p a, b q ÞÑ a ‘ b : “ $’&’% a ` b if a ` b P r´
M, M s M if a ` b ą M ´ M if a ` b ă ´ M , where ` denotes the usual sum in R ;(iv) b : A ´ M,M ˆ A ´ M,M Ñ A ´ M,M , p a, b q ÞÑ a b b : “ $’&’% a ˆ b if a ˆ b P r´
M, M s M if a ˆ b ą M ´ M if a ˆ b ă ´ M , where ˆ denotes the usual multiplication in R .If we further ask that A ´ M,M is closed under products and arbitrary summations, alongthe lines of what we did for A M , we get that A ´ M,M “ r´
M, M s , for all M ě . Noticealso that from the definition of addition ‘ it follows immediately that ‘ is commutative, but A ´ M,M “ p A ´ M,M , ‘ , b , ď A ´ M,M q is not a semiring. Indeed, addition ‘ is not associative. Aneasy counterexample is the following: ´ ‘ p M ‘ q “ ´ ‘ M “ M ´ , while p´ ‘ M q ‘ “´ ‘ M “ M ´ . n three classes of Non-Diophantine arithmetics t A ´ M,M u M ě can still be useful. Any A ´ M ,M still solvesthe paradox of the heap. If we then consider the functions ‚ s : A ´ M,M Ñ R , a ÞÑ s p a q ” a , so s is the identity function Id | A ´ M,M ; ‚ t : R Ñ A ´ M,M , a ÞÑ t p a q : “ $’&’% a if a P r´
M, M s M if a ą M ´ M if a ă ´ M ,it is immediate to see that, for all M , A ´ M ,M is weakly projective with respect to the realarithmetic R “ p R , ` , ˆ , ď R q , with projector s and coprojector t . Proposition 3.
Pick any real M ě , and any sequence p x n q P R N . For every n , define y n : “ ř n ď k x n . Then, the weak projection of p y n q converges in A ´ M,M if and only if one ofthe following holds:(i) lim inf n y n ě M ;(ii) lim y n exists and belongs to p´ M, M q ;(iii) lim sup n y n ď ´ M .Proof. The fact that t is continuous is obvious. Suppose then that lim n t p y n q “ t p lim n y n q Pp´ M, M q , where the equality comes from the continuity of t . By definition of t , we havethat either lim n y n belongs to p´ M, M q , verifying condition (ii), or it does not. In this lattercase, it may be that either lim n y n ě M , verifying condition (i), or lim n y n ď ´ M , verifyingcondition (iii).The fact that condition (ii) implies lim n t p y n q P p´ M, M q is immediate by the definition offunction t . Suppose now that lim inf n y n “ 8 ą M . This implies that lim n y n “ 8 . But then, lim n t p y n q “ t p lim n y n q “ M , where the first equality comes form the continuity of t , and thelast equality from the definition of t . A similar argument shows that if lim sup n y n “ ´8 ,then lim n t p y n q “ ´ M . If lim inf n y n “ L P r M, , the previous argument shows that lim n t p y n q “ M , while if lim sup n y n “ ´ L P p´8 , ´ M s , the previous argument shows that lim n t p y n q “ ´ M . This concludes the proof. (cid:3) We have a corollary to Proposition 3.
Corollary 1.
Pick any real M ě . The weak projection of any series of elements of R isabsolutely convergent in A ´ M,M . In addition, the weak projection of any series of elementsof R that is either convergent or divergent (to `8 or ´8 ), converges in A ´ M,M .Proof.
Immediate from Proposition 3. (cid:3)
Notice that, because addition ‘ is not associative, Lemma 1 does not hold in this class ofabstract prearithmetics. This means that t ˜ ÿ n “ a n ¸ ‰ t ˜ k ÿ n “ a n ¸ ‘ t ˜ ÿ n “ k ` a n ¸ . Hence, we need to project the entire series in the exact order we want the elements to besummed, otherwise Proposition 3 may not hold.There is a tradeoff in using this class instead of t A M u . We still manage to resolve theparadox of the heap, and we further show that the weak projection of series that diverge M. Caprio, A. Aveni and
S. Mukherjee also to ´8 , converges in A ´ M,M , for all M . The shortcoming, however, is that we loseassociativity, so working with elements of A ´ M,M may be difficult. Ultimately, the choice ofone or the other will depend on the application the scholar has in mind.4.
Class t B M u M P R ` . In this section we present a class of abstract prearithmetics t B M u M P R ` where every elementis a complete totally ordered semiring, and such that the projection of a convergent ordivergent series (to `8 or ´8 ) converges. There is a drawback: it does not resolve theparadox of the heap. Once more, the choice of using it will depend on the application thescholar has in mind.For every M P R ` , define B M “ p B M , ` , ˆ , ď B M q , where B M “ r , M s . We have that ď B M is equivalent to ď R , a ` b : “ M ˆ exp ` log ` aM ´ a ˘ ` log ` bM ´ b ˘˘ ` exp ` log ` aM ´ a ˘ ` log ` bM ´ b ˘˘ , and a ˆ b : “ M ˆ exp ` log ` aM ´ a ˘ ˆ log ` bM ´ b ˘˘ ` exp ` log ` aM ´ a ˘ ˆ log ` bM ´ b ˘˘ . Notice that we do not require to “force" and M to be the minimal and maximal elements,respectively; they come naturally from the way addition ` and multiplication ˆ are defined.In addition, we see that a ` b “ f p f ´ p a q ` f ´ p b qq , a ˆ b “ f p f ´ p a q ˆ f ´ p b qq , where f : R Ñ r , M s , f ÞÑ f p x q : “ M ˆ e x ` e x . The inverse is given by f ´ : r , M s Ñ R , x ÞÑ f ´ p x q : “ log ` xM ´ x ˘ . Hence, it turns out that B M is a projective prearithmetic, andthat its generator induces an isomorphism between R and p , M q . This tells us immediatelythat p B M , ` , ˆ , ď B M q is a complete totally ordered semiring, so addition ` and multiplication ˆ are associative, and that `8 and ´8 in R : “ R Y t8u Y t´8u correspond to M and ,respectively. This last particular tells us that the projection f p ř n a n q of any series ř n a n in R converges in B M , as long as ř n a n does not oscillate. The drawback to using B M is thatthe paradox of the heap is not resolved: there is no element in B M such that a ` a “ a . References [1] Diederik Aerts, Marek Czachor, and Maciej Kuna. Fourier transforms on Cantor sets: A study in non-Diophantine arithmetic and calculus.
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Department of Statistical Science, Duke University, 214 Old Chemistry, Durham, NC27708-0251
Email address : [email protected] URL : https://mc6034.wixsite.com/caprio Department of Statistical Science, Duke University, 214 Old Chemistry, Durham, NC27708-0251
Email address : [email protected] URL : Department of Statistical Science, Mathematics, Computer Science, and Biostatistics &Bioinformatics, Duke University, Durham, NC 27708-0251
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