IJETST - Vol.||03||Issue||02||Pages 3601-3607||February||ISSN 2348-9480 2016

International Journal of Emerging Trends in Science and Technology

Impact Factor: 2.838 DOI: http://dx.doi.org/10.18535/ijetst/v3i02.18 A Comparison Study of Non-Standard Analysis and Non-Archimedean Ultrametric Theory

Author Anuja Ray Chaudhuri Department of Mathematics, Ananda Chandra College Jalpaiguri, West Bengal, India, 735101 Email: [email protected] Abstract elements are mainly considered in non-standard analysis. Recently a new scale free analysis has been developed using the concept of relative , scale free infinitesimals and their corresponding non-archimedean absolute value. With this valuation the system R has been extended to an infinite dimensional non-archimedean system R accommodating infinitely small and large numbers. In this paper we present a comparison study of non-standard analysis and non-archimedean ultrametric theory. Key words: Hyper Real, Non-archimedean absolute value, Relative Infinitesimal, Scale Invariance.

1. Introduction Since the last few years we have been developing Non-Standard analysis is a branch of classical a Scale Invariant Analysis [3,4,5] on the of real analysis that formulates analysis using a rigorous numbers R using the concept of relative notion of an infinitesimal number. Non-Standard infinitesimals, scale free infinitesimals and their analysis was introduced in the early of 1960s by corresponding absolute value and applying this the mathematician . Much of formalism we extend R into an infinite the earliest development of infinitesimal dimensional metric space R which is a field by Newton and Leibnitz was formulated using extension of the set of rational numbers under the expressions such as infinitesimal number and new non-Archimedean absolute value [4]. vanishing quantity. These formulations were widely criticized by George Berkley and others. It 2.1 Approach to Non-Standard Analysis was a challenge to develop a consistent theory of There are two different approaches to non- analysis using infinitesimals and the first person to standard analysis: the semantic or model theoretic do this in a satisfactory way was Abraham approach and the syntactic approach. Both of Robinson [1]. In 1958 Schmieden and Laugwitz these approaches apply to other areas of proposed a construction of a ring containing mathematics beyond analysis, including number infinitesimals [2]. The ring was constructed from theory, algebra, topology and etc. sequences of real numbers. Two sequences were Robinson’s original formulation of non-standard considered equivalent if they differed only in a analysis falls into the category of semantic finite number of elements. Arithmetic operations approach. As developed by him in his papers, it is were defined element wise. However, the ring based on studying models (in particular saturated constructed in this way contains zero divisors and models) of a theory. Since Robinson’s work first thus cannot be a field. appeared, a simpler semantic approach (due to

Anuja Ray Chaudhuri www.ijetst.in Page 3601

IJETST - Vol.||03||Issue||02||Pages 3601-3607||February||ISSN 2348-9480 2016

Elias Zakon) has been developed using purely set- R is isomorphic to a subset of *R, since one can theoretic object called super structures. In this identify any real  R with the class of approach a model of a theory is replaced by an sequences { ……. }. Moreover *R is an object called a super structure V(S) over a set S. ordered field. Starting from a super structure V(S) one construct The syntactic approach requires much less another object *V(S) using the ultra power and to understand and use. This construction together with a mapping approach was developed in the mid 1970s by the V(S)→*V(S) which satisfies the transfer mathematician Edward Nelson. Nelson introduced principles. The map ‘*’ relates formal properties an entirely axiomatic formulation of non-standard of V(S) and *V(S). Moreover it is possible to analysis that he called Internal (IST) consider a simplified form of saturation called [6]. IST is an extension of Zermelo Fraenkel Set countable saturation. This simplified approach is Theory (ZST). Along with the basic binary also more suitable for use by mathematicians who membership relation, it introduces a new predicate are not specialist in model theory or logic. standard which can be applied to elements of the Let us briefly recall the ultra power construction mathematical universe together with some axioms of Robinson. Though less direct than the of reasoning with this new predicate. axiomatic approach, it allows one to get a more Syntactic non-standard analysis requires a great intuitive contact with the origin of the new deal of care in applying the principle of set structure. Indeed the new infinite and infinitesimal formation which mathematicians usually take for numbers are formulated as equivalence classes of granted. As Nelson pointed out, a common fallacy sequences of real numbers, in a way quite similar in reasoning in IST is that of illegal set formation. to the construction of the set of real numbers R For instance, there is no set in IST whose from rationals. elements are precisely the standard integers (here Let N be the set of natural numbers. A non- standard is understood in the sense of new principal (free) ultra filter U on N is defined as predicate). To avoid illegal set formation, one follows: must only use predicates of Zermelo-Frankel- U is a non empty set of subsets of N [P(N)  U  Choice (ZFC) to define subsets [6]. φ ], such that I. φ  U 2.2. Basic Definitions and Constructions of II. A U and B  U ⇒ A  B  U Extended Number Systems III. A  U and B  P(N) and B  A⇒ B  U An infinitesimals is a number that is smaller than IV. B  P(N) ⇒ either B U or { j  N : j  every positive real number and is larger than U} U, but not both. every negative real number, or, equivalently, in V. B P(N) and B is finite ⇒ B U . absolute value it is smaller than 1/m for all m  N Then the set *R is defined as the set of ={1,2,3,…..}. Zero is the only real number that at equivalence classes of all sequences of real the same time an infinitesimal, so that the non- numbers modulo the equivalence relation: , zero infinitesimals do not occur in standard analysis. Yet, they can be treated in much the provided {j : }  U, and being two same way as can be for the ordinary numbers. For sequences { } and { }. example, each non- zero infinitesimal Є can be Similarly, a given relation is said to hold between inverted and the result is the number ω = 1/Є. It elements of *R if it holds term wise for a set of follows that |ω| > for all  N, for which indices which belongs to the ultra filter. For reason ω is called hyper large or infinitely large. example : ⇒ { j : }  U. Hyper large numbers too do not occur in ordinary analysis, but nevertheless can be treated like

Anuja Ray Chaudhuri www.ijetst.in Page 3602

IJETST - Vol.||03||Issue||02||Pages 3601-3607||February||ISSN 2348-9480 2016

ordinary numbers. If for example, ω is positive classical mathematics. Their * transforms are hyper large, we can compute ω/2, ω-1, ω+1, 2ω indicated by *N, *Z, *Q, *X respectively. etc. The positive hyper large numbers must not be Selecting all finite numbers from *N and *Z we confused with , which should not be regarded a obtain again N, Z, but this is not true for *Q, number at all. simply because *Q (just as *R) contains finite If Є is hyper small, if δ is too hyper small but non non-standard numbers. But again there is a zero and if ω is positive hyper large, so that –ω is distinct difference between *Q and *R in this negative hyper large, we write Є  0, δ  0, ω  respect; there are finite elements t of *Q that , -ω  - respectively. It would be wrong of cannot be written as t = + Є, with  Q, Є  course, to deduce from ω   that the difference *Q, Є  0. For let c be any irrational number, say

between ω and  would be hyper small. c = , and let { } be some Cauchy Given any R,  0 and any δ  0, let t = + δ, sequence of rationals converging to c. The then Є < |t| < ω, for all Є  0 and all ω  . The sequence { } generates an number t is called finite (or infinitesimals δ in *R (because this sequence appreciable/moderately small or large) number (as converges to zero). On the other hand } it is not too small or not too large). generates an element  *Q ⊂ *R and is finite, Three non overlapping sets of numbers (old or but it has no standard part in Q, for otherwise new) can now be formed. Є for some  Q and Є  *Q, Є  0.

a) The set of all infinitesimals, to which zero But { } also generates the finite belongs. number  R, so that = δ  0. It b) The set of all finite numbers, to which all follows that =  0, hence =0 non zero real numbers belong. (as is ordinary real), which would mean that c) The set of all hyper large numbers, c  Q, a contradiction. containing no ordinary numbers at all. There are various ways to introduce new numbers. Together these three sets, constitute the set of all One way is done by means of infinite sequences numbers of “Real Non-Standard Analysis”. This of real numbers. In particular, the elements of *R set, which clearly an extension of R, is indicated will be generated by means of infinite sequences by *R and is called the * transform of R. The of reals and it will be necessary to consider all elements of *R are called hyper real. such sequences. (Recall that the elements of R can If a number is not hyper large it is called finite or be generated by means of rather special infinite limited. Clearly, t  *R is finite iff t = + Є for sequence of rationals, i.e., the Cauchy sequences). some  R and Є  0. Given such a t, both More generally, given any set X the elements of and Є are unique, for + Є = + δ, Є R, its * transform *X will be generated by means of Є, δ  0, we have (as  R) and   infinite sequence of elements of X, quotionted by δ. the equivalence class generated by the chosen By definition is called standard part of t, and ultra filter and again all such sequences must be this is written as = st(t). taken into account. For example {1,2,3,…….} The st provides an generates a hyper large element of *N, and important bridge between the finite numbers of {3/2,5/4,9/8,…….} generates a finite element of non-standard analysis and ordinary real numbers. *Q, and an infinitesimal, generated by Trivially, if t is itself an ordinary real number, {1/2,1/4,1/8,……}. Different sequences may then st(t) = t. generate the same elements of *X. In fact, given The * transform not only can be obtained for R any  *X there are many (uncountably many) but also for N, Z, Q and in fact for any set X of different sequences which form an equivalence

Anuja Ray Chaudhuri www.ijetst.in Page 3603

IJETST - Vol.||03||Issue||02||Pages 3601-3607||February||ISSN 2348-9480 2016

class under the ultra filter which represents the , then for some  element  *X. As a consequence, changing [ ]. finitely many terms of a generating sequence has A non-standard proof of this theorem proceeds as no effect on the element generated. follows : 2.3 The Purpose of Non-Standard Analysis Let,  *N be hyper large. Divide [ ] into Starting from N, the sets Z, Q and R have been equal subintervals, each of length introduced in classical analysis in order to enrich δ = ( )/ . Then δ  0. Let, be the smallest analysis with more tools and to refine existing element of *N such that * > 0, then tools. The introduction of negative numbers, of *  0. fractions and of irrational numbers is felt as a Let c = st( ), then by continuity, strong necessity, and without it mathematics and would only be a small portion of what it actually for certain infinitesimals is. The introduction of *N, *Z, *Q and *R, and . Hence  . however was not meant at all to enrich But  R, so that = 0. mathematical analysis (at least not when it all Terrence Tao, one of the most brilliant started), but only to simplify it. In fact, definitions contemporary mathematicians, has been and theorems of classical analysis generally are advocating strongly the use of non-standard greatly simplified in the context of non-standard analysis as soft analysis rather than using only the analysis. Non-standard analysis has also been classical hard analysis in particular differential applied later in a more traditional way, namely to equations and various other fields of applications introduce new mathematical notions and models. in his blog page ‘ What’s New’. Examples can be found in probability theory, So far we have discussed about Robinson’s asymptotic analysis, mathematical physics, analysis. Our approach is quite different and Economics etc. independent of conventional Non-Standard As an example of a simpler definition, consider analysis. In our work we have introduced an continuity. A function from R to R is continuous infinite dimensional metric space R which is an at c  R if statement (i) holds: extension of R using the concept of non-  Є  R, Є > 0:  δ  R, δ > 0 :   R, archimedean absolute value. | | < δ : | | < Є ………...……..(i) Now to there corresponds a unique function * , 3. Non-Archimedean Ultra metric Theory called the * transform of , that is a function from An absolute value on K is a function |.| : K→ *R to *R, such that * if  R and (i) that satisfies the following conditions : is true iff (ii), which is the * transform of (i) is | | = 0 iff = 0, true: | | = | || | for all  K,  Є  *R, Є > 0 :  δ  *R, δ > 0 :   *R, | |  | | + | | for all  K, | | < δ : |* | < Є …...…….(ii) We shall say an absolute value of K is non- Moreover (i) is equivalent to much simpler archimedean if it satisfies the additional condition: statement (iii) | |  max (| |, | |) for all  K  δ  *R, δ  0 :  ……(iii) Otherwise, the absolute value is archimedean. An illustration of a simpler proof is that of the Example- If we take, Intermediate Value Theorem: | | = 1, if  0 and | |= 0, if = 0 for any field If : R → R is continuous in the closed interval K, then it is trivially a non-archimedean absolute [ ], , and both finite, and value.

Anuja Ray Chaudhuri www.ijetst.in Page 3604

IJETST - Vol.||03||Issue||02||Pages 3601-3607||February||ISSN 2348-9480 2016

3.1 Non-Archimedean model between and (0,δ) ⊂ R, which we can write as

( ) =  ( ) for an infinitesimal   and a Let, *R be a non-standard extension of the real number system R. Let, 0 denote the set of relative infinitesimal 0 < < δ, . This may infinitesimals in R. Then an element of *R, be interpreted as by saying that for each arbitrarily denoted as x, is written as x = x+, xR and 0. small δ > 0, there exists in the non-standard *R an

The set 0 and hence *R is linearly ordered that infinitesimal   so that the dimensionless  matches with the ordering of R. The set 0 is thus equality of the form holds good  of cardinality c, the continuum. The non-zero independent of the scale δ. We, henceforth elements of 0 are new numbers added to R which identify with the set of relative infinitesimals are constructed from the ring S of sequences of in = (0,δ) ⊂ R so that ⊂ . We remark real numbers via a choice of an ultrafilter to that in this framework, a positive variable is remove the zero divisors of S. A non-standard defined relative to the scale δ by the condition infinitesimal is realized as an equivalence class of > δ. sequences under the ultrafilter and may be considered extraneous to R. The magnitude of an A relative infinitesimal  ⊂ = (-δ,δ) ( 0) element x of *R is evaluated using the usual is assigned with a new absolute value Euclidean absolute value .

, We now give a new construction relating ……..(II). We also set |0| = 0. It is easy to infinitesimals to arbitrarily small elements of R in a more intrinsic manner. The words ‘’ arbitrarily verify that is a non-archimedean absolute small elements’’ are made precise in a limiting value. sense in relation to a scale. Remark 1. We notice that there exists a non- Given an arbitrarily small positive real variable in trivial class of infinitesimals (those satisfying the sense that , there exists a rational  ) for which the value assigned to an infinitesimal a real number, i.e.,  δ. One of number δ > 0 and a set of positive reals satisfying 0 < < δ < and our aims is to point out the non-trivial influence of these infinitesimals in real analysis. This is to be the inversion rule contrasted with the conventional approach. The ……………………………….(I) where 0 < (δ) Euclidean value of an infinitesimal in Robinson’s << 1, is a real constant so that also satisfies the non-standard analysis is numerically an

scale invariant equation . infinitesimal.

The elements so defined are called relative Definition (I) is non-trivial in the sense that in infinitesimals relative to the scale δ. A relative absence of it, the scale δ can be chosen arbitrarily

infinitesimal is negative if - is a positive relative close to an infinitesimal , so letting , infinitesimal. The associated scale invariant one obtains = 0. Thus, dropping the inversion infinitesimals corresponding to the relative rule, we reproduce the ordinary real number system R with 0 being the only infinitesimal. infinitesimals is defined by .

Remark 2. An infinitesimal  has a Now because of linear ordering of , the set of countable number of different realizations, each positive infinitesimals of *R, that is inherited from for a specific choice of the scale δ, having R, and the fact that the cardinality of equals valuation . Indeed, given a decreasing that of R, there is a one-one correspondence sequence of (primary) scales so that as Anuja Ray Chaudhuri www.ijetst.in Page 3605

IJETST - Vol.||03||Issue||02||Pages 3601-3607||February||ISSN 2348-9480 2016

n→, the limit in the defn. II can instead be (iv) 0 is the union of at most of a countable evaluated over a sequence of secondary smaller family of clopen (both open and

scales of the form ,  for each fixed n. closed) balls. (v) The set 0 equipped with the above This observation allows one to extend that absolute value is totally disconnected. definition slightly which is now restated as Lemma 1. A closed ball in 0 is both complete and i) Scale free (invariant) infinitesimals compact.

satisfying 0 < (and

Proof. The proof follows from the following = as →) are called ( positive observations. Given ε > 0, consider a closed ) scale-free δ-infinitesimals. By interval [a,b] ⊂ 0 (in the usual topology) such that

inversion, elements of > 1 are 0 < a < b < ε. The valuation realizes this closed called scale-free δ-. interval as an ultrametric (sub) space U of 0 which

ii) A relative (δ) infinitesimal ( 0)  is an union of at most of a countable family of is assigned with a new (δ dependent) clopen balls. absolute value = = Now we consider completeness. A sequence { }

 . (In this scale ⊂ U is Cauchy ⇔ → 0 free notation, all finite real numbers ⇔ → 0 ⟹ there exists N > 0 such are mapped to 1 ). that ) = ( ) for all n  N. Now since for a non-zero infinitesimal , the associated It is easy to verify that |.| defines a non- archimedean absolute value on 0. valuation is non-zero, it follows that →  U in the ultra metric in the sense that = The set 0 is uncountable and the absolute value as . Compactness is a consequence of the satisfies the stronger triangle inequality. fact that any sequence in U has a convergent

Accordingly the set 0 = { 0,  δ }, r=0,1,…… subsequence. Indeed, a sequence { } in U can may be said to acquire the structure of a cantor set not be divergent in the given ultra metric since 0 

like ultra metric space. The set 0 indeed is realized  1. as a set of nested circle in the ultra metric norm, when we order, with out Next we extend this non-archimedean structure of 0 on the whole of R. any loss of generality, . The ordinary 0 of R is replaced by this set of scale free Let, infinitesimals being the for a real number  R. For a finite  R i.e, equivalence class under the equivalence relation when  δ , we have || || = = . For an , where  means .  on the other hand || ||

while for an arbitrarily From the ultra metric property of 0 we can say that large (→ ) i.e, when , we define || || = | | which is evaluated with the scale δ  1/N. (i) Every open ball in 0 is closed and vice- We notice that the above absolute value awards versa. the real number system R a novel structure i.e, for (ii) Every point in a ball is centre of the ball. an arbitrarily small scale δ, numbers and (iii) Any two balls in 0 are either disjoint or satisfying and now are represented [4] one is contained in another. as and . The ultra metric space { R ,||.||} is denoted as R. From Lemma 1 it Anuja Ray Chaudhuri www.ijetst.in Page 3606

IJETST - Vol.||03||Issue||02||Pages 3601-3607||February||ISSN 2348-9480 2016

is clear that R is locally compact, complete metric Mathematical Society, Vol.83, space. In our paper [4] we have proved that this Number6,(1977). model R is realized as a completion of the field of 7. D.P.Datta and S.Raut, The arrow of time, the rational numbers under this new non- complexity and the scale free archimedean absolute value ||.||. analysis,Chaos, Soliton and fractals,28,(2006),581-589. 4. Conclusion 8. S.Raut and D.P.Datta, Analysis on a In this formalism we have presented a new Fractal set, Fractals 17,(2009),45-52. elementary proof of well known Prime Number 9. S.Raut and D.P.Datta, Nonarchimedean Theorem [4]. Also we have applied this analysis on Scale invariance and Cantor sets, Fractals a class of differential equations [10]. We report in 18,(2010),111-118. particular, some simple but non-trivial 10. A.Ray Chaudhuri, Nonlinear Scale applications of this non-linear formalism leading invariant Formalism and its Application to to emergence of complex non-linear structures Some Differential Equations, IOSR even from a linear differential system. It is also Journal of Mathematics (IOSR-JM), shown that anomalous mean square fluctuations Vol.10, Issue 4 Ver.iv (2014),27-37. can arise naturally from the ordinary diffusion equation interpreted scale invariantly in the present formalism endowing real numbers with a non-archimedean multiplicative structure [5].

References 1. A. Robinson, Non-Standard analysis, North-Holland Publishing Co., Amsterdam,1976. 2. Curt Schmieden and Detlef Laugwitz: Eine Erweiterung der Infinitesimalrechnung, Mathematische Zeitschrift 69(1958),1-39. 3. D.P.Datta, S.Raut and A,Ray Chaudhuri, Ultrametric Cantor sets and Growth of measure, P-adic Numbers, Ultrametric Analysis and Applications, Vol.3,No.1,(2011),7-22. 4. D.P.datta and A.Ray Chaudhuri, Scale free Analysis and Prime Number theorem, Fractals, 18,(2010),171-184. 5. D.P.Datta, S.Raut and A Ray Chaudhuri, Diffusion in a Class of Fractal Sets, International Journal of Applied Mathematics and Statistics, Vol.30; Issue No.6(2012),34-50. 6. Edward Nelson: theory: A New Approach to Non-Standard Analysis,Bulletin of the American

Anuja Ray Chaudhuri www.ijetst.in Page 3607