American Journal of Applied Sciences
Literature Reviews Nonstandard Analysis for Some Convergence Sequences Theories Via a Transfer Principle
1,2Jwahir-H-Gdal
1Department of Mathematics, College of Science, University of Bisha, Bisha, Kingdom Saudi Arabia 2Department of Mathematics, College of Science, Sudan University of Science and Technology, Khartoum, Sudan
Article history Abstract: The purpose of this article to introduce nonstandard models for a Received: 03-11-2020 familiar types of convergent sequences theories through a transfer principle. Revised: 06-02-2021 The nonstandard analysis principle discuss, that any statements on areal Accepted: 08-02-2021 system can be extended to a similar structure over larger hyperreal system
Email: [email protected] therefore the results that hold true on the original system, remains true in a hyperreal system if and only if its ∗-transform is true. We apply such technique to transfer a classical proof for real sequences theorems, therefore we obtain an equivalents nonstandard proof on hyperreal system which is often clear, shorter and uncoblicated.
Keywords: Nonstandard Analysis, Hyperreal Numbers, ∗-Transform A Transfer Principle, Convergence Sequences
Introduction There were a number of studies have examined a results on nonstandard analysis and its applications. Sun Nonstandard analysis is anew mathematical (2015) were introduced an applications economics, technique that widely use in dever field of mathematics Similary (Duanmu, 2018) were applied the nonstandard and other science such as Statistics and economics analysis to Markov processes and Statistical Decision (Goldblatt, 1998). It become a powerful, mathematical theory. An interesting application of a transfer principle tool in the 20th century for providing a new method in for continuations of real functions to Levi-Civita field order to formulating statements and proving theorems has been presented by (Bottazzi, 2018). A new approach which yelding for an enlarge view of the mathematical to nonstandard analysis has been presented by land scape (Goldbring, 2014; Goldblatt, 2012). (Abdeljalil, 2018), he proposed a very simple method in The first who has been succeeded to demonestrated practice to nonstandard analysis without using the a rigorous foundation for the use of infinitesimal and ultrafilter, (Ciurea, 2018) has constructed an approach infinite numbers in analysis (Lengyel, 1996), and for nonstandard analysis in a complete metric spaces. provided a basic concepts of nonstandard analysis Numerous studies have attempted to the framework for was Abraham Robinson during 1960 s. Therefore a working with infinitesimals theory developed by Abraham new large field of numbers known as a hyperreal Robinson and application for nonstandard analysis we system includes areal number system, infinite, reported them in (Mocanu et al., 2020; Sanders, 2019; Bell, infinitesimals numbers which are non-zero, infinitely 2019; de Jong, 2020; Ciurea, 2018; Goldbring and Walsh, larg and small numbers were constructed (Keisler, 2019; Katz and Polev, 2017; Robinson, 2016). 2000). In fact a hyperreal numbers system can be We apply nonstandard analysis concepts to give regarded as extension order field of areal numbers * nonstandard models equivalents for the classical theories areal number system infinitesimals. Moreovere A. of real sequences that converges. The method we used is Robinson has been showed that a relational structure a cording to Abraham Robinson for transferring over real numbers can be transferring to equivalent properties from areal number system to ahyperreal structure over hyperreal system (Hurd and Loeb, system * via a transfer principle therefore all properties 1985). Therefore every statement holds true within of sequences of the real numbers were preserved. The real system remain true in hyperreal system hence this equivalent proofs we obtained often simpler and directly propriety for transferring statements known as a which is desire result. transfer principle which based on Robinsons approach The study compose of five section we organized it as (a nonstandard analysis). follows:
© 2021 Jwahir-H-Gdal. This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license. Jwahir-H-Gdal / American Journal of Applied Sciences 2021, Volume 18: 33.50 DOI: 10.3844/ajassp.2021.33.50
The first section begin by giving a brief overview than every negative real number or equivalently in history of nonstandard analysis as introductory section. In 1 absolute value it is smaller than for all m0 section two show how a hyperreal system can be created, m0 as an order field extension of real numbers and as ultra- (Ponstein, 1975). Although zero is only infinitesimal power. A notations about a infinitesimal numbers and an number which belong to real number system, it may be ultra-power construction of the order field for the extended in some way in order to include all infinitesimals hyperreal system * were introduced. (Ponstein, 1975). The following proposition illustrates In section three and four, we present an introduction to some facts in a hyper areal numbers . idea of theory for the language of relational structure in A non zero number k on is said to be: order to describe a transferring properties. We formulate transfer principle and then introduce a Infinitely small or (infinitesimal) if, |k| 1 , for first-order sentences which the transfer principle applies m0 to by replacing statements by its ∗-transform. We the every integer m0 logical construction used in essentially way to transfer a Finite if |k| m for some m statements from standard form into a nonstandard model. 0 0 1 The final section draws upon the entire study, we Hyper larger or (infinitely large) if m0 , >m0, present a familiar concepts of convergence sequence k theorem including, convergence theorem, limits theorem, for all m0 cluster point, Cauchy sequence and a monotonic A hyperreal numbers b is said to be positive sequence theorem. We discuss their nonstandard model which equivalents to its standard formulas. infinitesimal if b > 0 but less than a number o < a , negative infinitesimal if b < 0 but its greater than a The Structure of the Hyperreal Numbers negative number o < a . A hyperreal numbers is said to be infinitesimal if it either positive or negative System infinitesimal or zero (Goldblatt, 2012; Keisler, 1976). The hyperreal number can be constructed within Hens the sets of all infinitesimals numbers to which zero two approach, the first one as an extension order field belong and the set of all hyperlarg numbers which of a set of real numbers, and the other as ultra-power. containing a classical numbers all together are constitute a The two approaches allows us to extend arbitrary nonstandard numbers therefore they are clearly an functions and relations from to * . extension of the real numbers . Hens the large system of numbers which contains The Structure of the Hyperreal Number System as areal numbers system, infinite, infinitesimals numbers Complete Order Field that are non-zero, known as nonstandard or hyperreal numbers and denoted by * . A real numbers system is a complete order field ,,.,0 where +, and < are the usual algebraic Limited, Unlimited and Appreciable Numbers operation (relations of addition, multiplication and linear Suppose b * and s, t then the following ordering) on . Recall that real system can be extend to an propositions are holds, orderd field denoted by which contains an isometric copy of but strictly larger called a hyperreal or Proposition. 2.3.1. nonstandard system (Staunton, 2013; Hurd and Loeb, A number b is limited if s < b < s for some numbers 1985). Furthermore the construction of hyperreal number is s, t . reminiscent of the construction of the reals from the rationales numbers by means of equivalence classes of Proposition. 2.3.2. Cauchy sequence (Hurd and Loeb, 1985) (Staunton, 2013). In fact an elements of hyeperreal number * should be A number b is positive unlimited if s < b for all s . viewed as infinite sequence of real numbers. For example, Proposition. 2.3.3. the sequence 1,2,3…… should represent some infinite element of . However, many different sequences of A number b is negative unlimited if b < s for all s . rational numbers represent the same real number . Proposition. 2.3.4. Now we will introduce a basic arithmetical stricter of a hyper real and its relation to areal numbers. A number b is unlimited if it is positive and negative. Number Systems and Infinitesimal Proposition. 2.3.5. An infinitesimal is known as a number that is smaller A number b is positive infinitesimal if 0 < b < s for in magnitude than any non-zero real number and is larger all 0 < s .
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Proposition. 2.3.6 Quotients
A number b is negative infinitesimal if s < b < 0 for a a c , and Infinitesimal all 0 > s . c e e c Proposition. 2.3.7 , c 0 Appreciable d A number b is appreciable s < |b| < t for some s, t . ce In facts all hyperreal numbers and an infinitesimals , , a 0 Unlimited aa are (finite). e , c 0 Unlimited Proposition. 2.3.1 c
The product of an infinitesimal and a finite number is Observe that the limited numbers and infinitesimals infinitesimal. are each a subring of * . An integer m is limited exactly when it is standard and if m is nonstandard it said to be illimited, for The Standard part of Hyperreal Numbers 1 example the rational m is limited without being The standard part play a basic role in infinitesimal x theorem, it connect between the finite numbers of 1 standard, moreover its true that the rational m 0 is nonstandard analysis and the classical numbers i.e., it x rounds off each finite hyperreal to the nearest real infinitesimal when n is illumined. Here is some useful number therefore infinite hyperreal numbers never rules which dealing with preceding notations: possess standard part.
Definition. 2.5.1 mislimitedegerint,0 minm 10. ii 1in Let s be finite (or a limited number) then the 1 unique real number y that is infinitesimally close to s, (s If m = and x 0 then m 0, i.e., m is infinite x t) is called the standard part (or shadow) of x. we denote it simal. by st(s) or sh(s). Thus we have t sh(s) or (t sh(sx)). 1 If and only if is unlimited. Theorem. 2.5.1 x The terms finite and infinite are often used for limited Every limited hyperreal numbers s is infinitely and unlimited. Notice that b is limited for some n , closed to exactly one real number called the standard unlimited iff for all, |b|
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Definition. 2.7.1. (The Monad Set) m(x) = {yR: y = x + z, zm(0)} G(x) = {yR: y = x + z, zG(0)} Let x be a hyperreal number, the monad of x is the set, denoted by: Corollary. 2.7.1 monxyxy : The quotient field G(0)/m(0) is isomorphic to the standard field of (Andres and Rayo, 2015). Definition. 2.7.2. (The Galaxy Set) Proof The galaxy of a set x is the set: m(0) is the kernel of the linear (over ) map st, i.e.:
glaxyxyisfinite :. mxxGstx 0:0.
Remark The following theorem explain existence and I. The monad(0) is the set of infinitesimals and uniqueness prosperity of standard part. gala(0) is the set of finite hyperreal numbers. II. Any two monads are equal or disjoint. The Standard Part Principle and the Mapping III. Let be two monads m(x) and m(y) be two monads Theorem. 2.8.1. (Standard Part Principle) then they are ether: Every finite number x is infinitely close to a Equal if x y unique real number y . Therefore every finite Disjoint if x y) monad contains a uniquely number on .
Proposition. 2.7.3 Proof The relation is an equivalence relation on * . Let x be in is infinitely close to a unique real number y . Then every finite hyperreal number x is Proof infinitely close to a unique real number. The relation equivalence relation if it satisfies the Uniqueness following condition. For any finite hyperreal x, y, z , we have: Consider y, z and y x, z x. Since is an equivalence relation we have y z, hence y-z 0. But y-z I. x-x 0 is in * . is in , so y-z = 0 and y = z. II. x-y y-x, so x y implies y x III. if x-y and y-z, in * then x-z is * Existences Let E = {y : y < x} be a nonempty set. A set E Thus from I, II, III then () is an equivalence relation has an upper bound if there is real number y > 0 such * on . that: |x| < y, whence –y < x < y so, -y E hence y is an Theorem 2.7.1 upper bound of E. Since is complete ordered field, so the set E has at least upper bound h. For every y The set monad (0) of infinitesimal elements is a where y > 0 we have: * subring of and an ideal in galaxy (0). That is: yxhy , (i). Sums, differences and products of infinitesimals are infinitesimal (ii). The product of an infinitesimal and a finite element It follows that x-h ≃ 0. Hence x ≃ h (Keisler, 1976). is infinitesimal Definition. 2.8.2. (The Standard Part Map.) (For proof the see theorem (1.4) (Keisler, 1976)) The map st: xst(x) called the standard part map. Proposition 2.7.4 Clearly maps x onto st(x) since st(x) = x, when x , I. Two galaxies G(x) and G(y) are either equal if x-y is furthermore it preserves algebraic structure as in the finite or disjoint following theorem (Andres and Rayo, 2015). II. If x 0 then m(x) is a translate of m(0) Theorem 2.8.3
Therefore for every x then: For every x, y :
36 Jwahir-H-Gdal / American Journal of Applied Sciences 2021, Volume 18: 33.50 DOI: 10.3844/ajassp.2021.33.50 i. st(x y) = st(x) st(y) the hyperreals * , we must construct a hyperreal number ii. st(x.y) = st(x).st(y) system * as an ultrapower of the real number system iii. x y implies st(x) st(y) (Hurd and Loeb, 1985). To present an equivalence relation iv. st(|x|) = |st(x)| we need the notion of an ultrafilter to do so, first we must v. x < y implies st(x) = st(y), iff x-y or (x ≃ y) present a definition of a filter (Staunton, 2013). vi. st(max(x, y)) = max(st(x), st(y)) Definition. 2.3.1. A Filter vii. If a , then st(a*) = a Let be a nonempty set of . A filter on is a The existence of standard of limited numbers follows nonempty collection of having the following from the Dedekind completeness of areal numbers . In properties (Staunton, 2013): fact the existence of standard part is a tentative formulation of completeness. (i). The empty set Theorem. 2.8.4 (ii). If U, V then UU (iii). If U and VU, then is cofinite or V Every limited hyperreal x is infinitely close to exactly one real number implies the completeness of Definition. 2.3.2. A Free filter (Goldblatt, 2012)[ theorem ( 5.8.1)]. In the following section is we will show the If all elements of a filter are infinite sets then it said constructing of a hyperreal number system as linearly to be free (or non-principal). ordered field based on the ultra powers construction of nonstandard model. Proposition. 2.3.1. A Construction of Hyper Real Number System as a. Every filter contains the nonempty set Ultra-Power b. = {} is smallest filete on c. A filter is apropper if The construction of hyperreal number from the a d. A filters are closed under finite intersection real number is similar to construction of a real from the rational numbers by means of equivalence class of Definition 2.3.3. An Ultra Filter Cauchy sequences (Staunton, 2013). To construct a A filter is said to be an ultrafilter denoted by iff any hyperreals * , first we illustrate some notion of an subset U of either U or Uc (not both by(i), (ii)) where: ultrafilter, which will allow us to do a typical ultra power construction of the hyperreal numbers. c UU Suppose for = {1,2,…..} be the set of real- valued sequences, under point wise addition and If is an infinite set the collection y = {AI: I-A is multiplication. Let u = (u ), v = (v ) are elements in l i i finite} is a filter called the cojnite or Frichet filter on . which defined as follows: Proposition. 2.3.2 i. uv = uivi, i Let U, V and Uc be a complement of U then the ii. u⨀v = u ⨀v , i i i following properties are holds:
However, uv and u⨀v are in , so its is closed i. UV, UV iff U, V under point wise addition and multiplication. So ( , , ii. Uc iff U ⨀) is a commutative ring with identity sequence, iii. An ultraflter is an ultrafilter on iff is however, satisfies all the properties of a field with amaximal proper filter identity, (1,11,1,…) and (0,0,0,…) = 0 and additive inverse consider, for example, the two sequences u = Definition. 2.3.4. The Fre’chet Filter (0,1,0,1,…), v = (1,0,1,0…) neither of u or v equal to the zero. However, point wise multiplication would give us: If is an infinite set then collection:
uv. 0,1,0,1,... 1,0,1,0... 0. U : U is finite
Thus two nonzero elements u, v whose product is zero, is called a Fre’chet filter or cognate. are prevent the sequence to be an order field. To The Fre’chet filter is not an ultrafilter Moreover its avoid this problem and to introduce equivalent relation proper iff is infinite. An ultrafilter on is free if it which make into an order field and then ex tend it to contains . Hence a nonprincapl ultrafilter must contain all
37 Jwahir-H-Gdal / American Journal of Applied Sciences 2021, Volume 18: 33.50 DOI: 10.3844/ajassp.2021.33.50 a finite sets. This is a critical property used in construction Elements of M are called nonstandard or hyperreal infinitesimals and infinitely large numbers. Here are some number s and technically its known as an ultrapower important properties of (Staunton, (2013). (Goldblatt, 2012). Definition.2.3.5. Free Ultra Filter Lemma. 2.5.1 Combining definitions (2.3.3) and (2.3.4) we come up The relation is an equivalence relation on a with the definition of a free ultrafilter. hyperreal . Also is said to be free if it contains the Fre’chet filter. Let M denote the set of all equivalence classes of * A free ultrafilter on contain any finite set of . in duced by . The equivalence class containing a Moreover, all elements of the are infinite sets particular sequence u = (ui) is denoted by [u] or u. Thus (Davis, 2009). if u v in * then u = [v] = [u] = u. Elements of are The free ultrafilters doesn’t always exist. Hence, an called nonstandard or hyperreal numbers. We use the ultrafilters are important for the purpose of construction same idea to define operation u v and relations which * of hyperreal . make into an ordered field. An infinite sequences of real numbers are represent a Equivalence class of a sequences. u, v under free ultrafilter often use to give a rules for equality and relation denoted by [u], [v] respectively thus: identification, so we can come up with a mathematically consistent and sensible system of hyperreal numbers therefore in this way a hyperreal number can be generated. uvuv :
An Equivalence Relation on Real Valued Sequence vuvv :
The relation on is called an equivalence We now define an operations and relations which we relation if it is -is reflexive (u u), symmetric (u v v will used it to make into an ordered field. u) because is a symmetric relation on and transitive (u = v and v = w imply u = w) because of Definition. 2.5.1 conditions (ii) and (iii) for a filter. The equivalence Let u = [u ] and v = [v ] then: relations are use the notation u v. i i
The set can be divided into disjoint subsets i. u + v = [u + v ], i.e., [u] + [v] = [uv] (called equivalence classes) by the relation . Each i i ii. u.v = [u .v ], i.e., [u].[v] = [u⨀v] equivalence class consists of all sequences equivalent to i i any given sequence in the class, therefore u and v are iii. [u] < [v] iff [u < v] iff {i : ui < vi} said to be in the same equivalence class iff u = v. Two sequences which differ at only a finite number of places So the equivalence class of the sequences of n-th are equivalent under (Hurd and Loeb, 1985). term is given by:
The Equivalence Classes on a Real Valued uvuvuviiiiii , Sequence and an Ultrapower uvuvuviiiiii ... In order to extend the real numbers system to the hyperreal * in an ultrapower concept we can use Not that the quotient set of under can denoted infinite sequences of real numbers (Davis, 2009) before * doing so, we shall create a field of real-valued sequences, by uu: . in which every standard real numbers are embedded as the corresponding constant sequence. Remarks Let M denote the set of all the quivalence classes of is technically known as an ultra-power. We in deuced by . The equivalence class containing a have used the same idea to define operations and particular sequence u = (ui) is denoted by [u] or u. Thus relations which make into an ordered field if u = v in d then u = [u] = [v] = v. (Goldblatt, 2012). We can define a relation on by putting: Logically for u, v we ca n replace the set, {r
: ui = vi} by ⟬u = v⟭ thus u v iff ⟬u = v⟭. u v ifandonlyif ii Now since we have all the necessary tools so that we
are ready to show that that * is order field. i:, uii v Theorem.2.5.1 When this relation holds it may be said that the sequences ui = vi possess same values at almost i. The structure is a linearly ordered field.
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Proof Thus, we have that is a totally ordered field. Hence a hyperreal numbers is totally orderd field numbers That * is a commutative ring with zero 0 = (Hurd and Loeb, 1985). [0,0,0,….] and unit 1 = [1,1,1,…..] and additive invers The filed is embaded into through a mapping given by -[ui: i ] = [-ui: i ] (i.e., if u 0 then there that assign to each u , the hyperreal u* denoted 1 1 is an element A, B, so that u.u = 1). To show that by the equivalence class of the constant sequence with it has multiplicative inverses, suppose [u] [0]. Then value u we shall identify u and u*. {i : ui = 0} and so {i : ui = 0}, since is Now we want show that areal number system can be ultrafilter, hence w = {iN: ui 0} is ultrafilter. Define embedded isomorphic as linearly ordered sub field of . 1 1 u = [ui] where [ui] = ui if ui 0 and [ui] = 0 i.e.: One can relate areal number u with constant sequence u = [u, u, u,……….] assign to the elements. 1 Define (u) = u* where, u* = [u] = [u, u, u,……….]. For u, , ifiw ui ui v we have the following properties: 0, otherwise * ** u v u v Then ⟬u⨀v = 1⟭ is equal to w so: u v u v* **
** uv1. u v iff u v ** u v iff u v Let r⨀s 1 then [u][v] = [u⨀v] = [1] is in hence [v] is multiplicative inverse of [u]1 or [u]. Now let A = Theorem. 2.5.2 ⟬u < v⟭, B = ⟬u = v⟭, C = ⟬v < u⟭ we want to show that The map : u u* is an order preserving field is a linearly ordered field with the ordering given by < . isomorphism from to * . Hence exactly one of: Proof uvuvuv ,, The mapping is one to one for u* = v* then, [u, u, u,……….] = [v, v, v,……….] and so u = v. To explain Is true. The element u of is positive if u > 0. We that a mapping *Preserves the field order properties we must show that the set {[u]: [0] < [u]} of positive must show that: elements in a hyperreal is closed under addition, multiplication and the law of Trichotomy which states i. [u, u, u,……….] + [v, v, v,……….] = [u + v, u + v, for a given element u either: u + v,……] ii. [u, u, u,……….].[v, v, v,……….] = [u.v, u.v,
uoruoru00,0 u.v,……….], establishes, (u + v)* = u* + v* for (i) and (u.v)* = u*.v* for (ii) respectively (where, -u is the additive inverse of u) (Law of trichotomy). To demonstrate suppose [u] = [ui] and define: The details are left to the readers. This results is useful for identify the real number u with u* to regard as Subfield of . In particular we LiuMiu :0ii ,:0 , may identify [0] with 0 and [1] with 1 (Hurd and Loeb, Niu :0i 1985; Goldblatt, 2012). Proposition. 2.5.3. We want to show that only one of k, m and n is in , from the law of trichotomy in , we see that k m n Given : * , we define a map on entities by = . Now one of l, m and n are in also kc, mc, nc G = {g: gG}, the map gives the embedded standard are in also: copy of G. The is superstructure homomorphism of if and only if it is a one to one map on . Hence the c c c c preserves the following, k m n k m n (read elemnt of): If G is entity of and G, then which is a contradiction. g*G* Clearly the fact that is totally ordered follows from = (equality): If G is an entity then {(u, u): uG}* = the fact that is totally ordered and that is an ultrafilter. {(v, v): vG*}
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* *** (Davis, 2009). We define the extension L* of a subset L A finite set: {g1, g2,…. gn} = g12 g, g ...... n for g1, of areal to be the set: g2,…. gn Set operations: = *, (GK)* = G* K*, (GK)* LuuLi** |, = G*K*, (G/K)* = G*\K*, (G K)* = G* K*. G, K i are entities Therefore L is the set of equivalence classes of The domain and ranger of n-aray relations and sequences whose values range over the elements of L. commutes with permutations of the variables. Then the a function can extend to a function * * * * (dom()) = dom( ), (rang()) = rang( ) therefore such extension processes same * Atomic standard definitions of set {(u, v): u, vG} rules as the original functions and relations. = {(c, w): v, wG*}.
The Formal Language of Relational Structure Moreover produces a proper extension G*G with equality iff G is a finite set. Those properties imply that In this section we will give an elementary idea on the image * is a nonstandard model in the formal logic the language theory of relational structure with a few sence which we will describe below. examples. We will use a formal logical symbol to express Remark statements that were asserted to be true or false of the Therefore we conclude the fact that: structure and .
The Simple Formal Languages for Relational Systems The * , * , * and are extensions of , , and the Logical Structure of First Order Statement and , respectively The hyperreal extension * preserves all order Definition. 3.1.1. (Relational Structure) properties of an ordered fields, hence a real numbers A relational structure is a system = {, q, f} form of a hyperreal numbers and the order relation. consists of a nonempty set , a collection of finitary Therefore is an ordered field extension of relations q on and a set f functions relations on .
The following principle is a necessary for extending Definition. 3.1.2 sequences and functions to the hyperreals (Davis, 2009). A language L is a set that including all logical The Extending Principle symbols and quantifiers (including the equality sign and the parenthesis) and some arbitrary number of constants, Each function in the standard models can be extended variables, function symbols and relation symbols it to a function acting on the corresponding nonstandard (Stroyan and Luxemburg, 1977). models. To be accurately for every real function f of one In order to describe a formal language it is first or more variables there is a corresponding hyperreal f * necessary to describe the symbols of the language and of the same numbers of variables, f * denoted the natural then we can describe the process of forming sentences. extension of f. each relational stricter then L is language L is bases in We can extend real-valued functions to hyperreal- the logical symbol. Therefore any statement that is valued functions in the following ways, let f : be expressible in logic structure is mentioning only standard a function then for every real-valued sequence , let numbers is true in if and only if it is true in . f(u1), f(u2),……… for ui , i then the extended The basic symbols is divided into two types: function f ***: by can defined as: (1). The symbols consists of logical symbols which are common to any simple language and do not vary if fufu* the statements is changed as containing the following symbols: for any hyperreal number [u] * . In other words: Logical connective: (and), (or), (inplies) (if f and only if), (not). With their usual fu* uf uf, .,,,,,... u 1 212 interpretation Quantifier symbols: (for all), (there exists) If there is a function f: L where L , to extend Operation, relation and function symbols, P, f, the function f to the hyperreals, we have to define the , et cetera extension of its domain L to a subset L of the hyperreal Parentheses and bracketes (,) and [,], ,
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An infinite list of constants which formed asset If is L-formula and x is any variables symbol of Symbols which usually denoted by Roman or and P is subset of Q then, (xP), (xP) are Greek letters L -formula The basic predicates are and = An infinite list of variable: Which is countable Now we will give some basic concepts in collection of symbols in which we use letters x, mathematical logic. y, z,…… An infinite list of function symbols: f, g, h, A Sentences and an Atomic Relation ……… Definition. 3.3.1. (A Sentences) An infinite list of relation symbols: P, R, Q, ………. A sentence known as a formula in which all variables Are bounded. If the closed terms of the (2). The symbols in the second category depend on Q and sentence are all defined then it has a fixed meaning will be called parameters. They consist of constant and it is either true or false. Symbols, Relation Symbols, function Symbols Definition. 3.3.2. (A Free Variables) The Formula and the θ-Term A free variable is a variable which obtained by There are expressions like composite functions in Replacing any variable occurring in a statement by some usual mathematical notation, constant, variable and constant to obtain another meaningful statement. function symbols is a string of symbols from the Definition. 3.3.4. (Bounded Variables) alphabet, they are special cases of terms we denoted L-term and they are defined inductively by the A variable that is not free is called a bounded or following laws: dummy variable.
Each constant symbol is an L- term. Definition. 3.3.5. (A Tomic Sentences)
i. Each variable symbol is an L- term. An atomic sentences is a formula which has no variables ii. If f is the name of a function of n variables and written are of the form Q(1,…,n), where Q is n-ary and 1,…,n are L-terms, then f(1,…,n) is relation and (1,…,n ) resents terms’ of the sentences. A an L-terms sentences also may defined inductively (Goldblatt, 2012). Definition 3.3.6. (A Simple Sentences) For example f(2, g(x, y)) and cos(x + y) are L- terms A simple sentences known as a language takes two Definition. 3.2.1. (A Closed Term) types denoted by, atomic and a compound sentence and A closed term is term which that made up of constant consists of a basic and combinations symbols which and functions symbols. It is undefined if it does not defined as a string of symbols in a sentence. name anything. An Atomic Relation Remark An atomic relations are simplest mathematical A term containing no variables is called a constant relations by which are meant relations containing neither term logical connectives nor quantifiers, hence relations such, A closed term is on that has no variables and as (=, ,…etc.) (Staunton, 2013). therefore made up of constants and functions Atomic relations can be regarded as functions be {true, symbols. A closed term is undefined if it does not false}. A sentences are of the form Q1,…,n where Q the i name any things. For example the constant names name of an n-ary relation and the (I = 1,…,n) are constant terms therefore 1,…,n are all closed i.e., there is no itself and f(1,…,n) is undefined if one of 1,…,n is undefined variables in the formula (Goldblatt, 1998). An arbitrary statement which composed of a finite Definition.3.2.2. (A formula) number of atomic relations and logical connectives can be define it inductively as follows, if and are two If , are L-formula then it follows that: sentence, then the following are also sentences: If is a formula so is ¬. If and ψ are formulas, then so is () If is a formula and x is a variable then (x) is also a formula
41 Jwahir-H-Gdal / American Journal of Applied Sciences 2021, Volume 18: 33.50 DOI: 10.3844/ajassp.2021.33.50
The Truth Value of Sentences Recall that a sentence is either true or false in the real (x) (x) number system. Let are two sentence with standard meaning of symbolic connectives , , , , we will Not that all the logical connectors can be derived present some rules that usually using for calculation a from logical symbols , , as follows: truth values of a sentences:
( ) = (()()) are true if are true and are true ( ) = ( ()) are true if are true ore are true ( ) = ( ()) (()) are true if are false (x) = (x) is true if and only if either is false or else
is true Example. 3.4.1 is true if and only if and either both true The first order field axioms can be expressed as first ore both false order logic statements as follows: The mathematical formulation for statements Associativity Prosperity: associated with truth values rules able use to distinguish
exactly which property is can transferable from to * xyzxyzxyz and vice- versa. xyzxyzxyz The Transfer Principle Commutative Prosperity: The transfer principle is the powerful tool that xyxyyx allows us to use the methods of non-standard analysis to prove results in standard analysis (Staunton, 2013). xyxyyx The transfer principle states that” a formula is true on areal system if and only if the corresponding formula is Distributive Prosperity: true on ".hence the transfer principle allow us to show that a hyperreal has all the properties of and also we xyzxyzxyx z can prove theorems about by first proving them in on the other words a transfer principle extends all a Existence of Identities: classical rules on a reals system to the hyperreal system
which allow for easier and more intuitively natural proofs x0 x Ay x y y in a hyperreal system (Davis, 2009). x 1 x Ay x y y The -Transforms for First-Order Sentences
Existence of Inverses: Definition 4.1.1
xxxy 0 The *-transform of a simple sentence Q in L-formula is the simple sentence Q* in -formula obtained by starring xxyxy 01 all function and relation symbols in the sentence Q.
Thus, constructing a ∗-transform of a sentence L Total ordering Property: really just consists of putting a ∗ on every term in Q, putting a ∗ on any relation symbol in Q and putting a ∗ xy x yy xx y on every set in Q acting as a bound on a variable (Davis, xyzx yy zx z 2009; Keisler, 1976; Goldblatt, 2012). yxyx y V y x Notation Note that the ∗-transform arises by attaching the We now will introduce the notion of the -transform prefix to symbols but not attaching to variable symbols of first-order L sentence which is useful tool for (Goldbring, 2014). transforming L-sentences in areal to the L * sentence First we introduce a number of example which in a hyperreal * . illustrate how we the a statements can be transforming.
42 Jwahir-H-Gdal / American Journal of Applied Sciences 2021, Volume 18: 33.50 DOI: 10.3844/ajassp.2021.33.50
Example. 4.1.1 Replacing each term occurring in Q by Q* Replacing the relation symbol g of any atomic Any positive real has areal n-th root for all n . formula occurring in Q by g* This statements can be formulated as: Replacing the symbol g of any quantifier (xP) or * n (xP) occurring in Q by g nxyxy . (1) The symbols <,>; will denote the corresponding relation and functions in in and (Goldbring, Which is true. We can transform it to the true 2014) sentence:
Thus, constructing a ∗-transform of a sentence L nxyxy*** n . (2) really just consists of putting a ∗ on every term in Q, putting a ∗ on any relation symbol in Q and putting a ∗ Which assert that, a hyperreal number has a hyperreal on every set in Q acting as a bound on a variable (Davis, n-th root for all n * . This the a ∗-transform of a 2009; Keisler, 1976; Goldblatt, 2012). For writing first- sentence (1) which is also true. order sentences, we can construct a method of transforming sentences in to sentences in as * Example. 4.1.2 follows, the *-transform of an L-term which obtained by replacing each function symbol f occurring There does not any numbers x . such that x<1. in by f *. Therefore we define ∗-transforms of sentences This statement can formulated as follows: explicitly on the construction of the sentence and xx 1 (3) inductively as follows:
()* := ()* Which is true. We can transform it to the true sentence: * * * () := * * * xx* 1 (4) () := * * * ( ) := * * * Which assert that there are no number x of * ( ) := * * * smaller than 1. (xQ) := (xQ) From This example we conclude that a number of (xQ)* := (xQ)** * / must be larger than all elements of , hence is infinitely large (unlimited). We now will an examples of first order L-formula and its equivalence ∗-transform Example 4.1.3 Example. 4.1.4 Archimedean property assert that, given any real numbers x there exists a natural number n (depending on The following first-order of L-formula of the totally x) such that x equivalent to first order L * as follows: Archimedean property can expressing as follows: xnxn (3) Associativity Prosperity: Which is true. The ∗-transform of (3) can gives by: x*** y z x y z x y z *** x ** n x n (4) x y z x yz xy z Therefore since n is unlimited, then the statement ( 4) Commutative Prosperity: is true in * . x ** y x y y x The ∗-transform of terms can be defined by induction on the formation of as following laws: x ** y x y y x * If is a constant or variable symbol then = . Distributive Prosperity: * * * * If = f(1,…,n) then = = f (1 ,…,n ) x *** y z A -transform of a sentences can be defined as follows: x y z x y x z 43 Jwahir-H-Gdal / American Journal of Applied Sciences 2021, Volume 18: 33.50 DOI: 10.3844/ajassp.2021.33.50 Existence of Identities: Proof We will prove this by induction on sentences. For xxyxyy**0 the base case, suppose the atomic formula P(1,…,n) ** xxyxyy1 is true for chosen values of 1,…,n. The -transform of this sentence is: Existence of Inverses: Pjj**,...... 11 ** x y x y 0 which is true if and only if the set of indices j is in x ** x 01 y x y ultrafilter. But the *-transforms of constants in are just the corresponding constant sequences, so by the definition * Total ordering Properity: of it follows that the set of indices j such that: ** ** Pjj ,...... xyxyyxxy 11 *** xyzxyyzxz are in , which must be in our ultrafilter. Therefore: yxyxyyx** * PP11,...,,...,.nn L So the list of first-order sentences * above are true in * . Conversely, suppose P(1,…,n). Then, by the same argument, set of indices j such that: Recall that the -symbol can droped in the following case: If the symbol refer to the transforms of well-known is the empty set, which cannot be in our ultrafilter. relations such as =, , <, , , ……etc. Therefore: If the symbol referring to well-known mathematical functions such as sin, cos, tan, cot, …….. etc. * PP11,..., nn ,..., If: XY, then f * : X *Y * and f *(x) = f (x) if xX. * Often the *- symbol in f may be dropped hence: Consider addition in . Its ∗-transform is ∗- addition in and x+y = x + y if x, y . The - * PP,..., ,..., symbol can safely be dropped 11nn Atomic relations are relations in which neither logical Hence the ∗-transforms of first-order sentences, connectives nor quantifiers play a part, but only such which we know to be true by the transfer principle relations as < or , etc. Consider first < in , leading to (Goldblatt, 1998). <* in . Similarly as under e) we have that x <* y is Thus the Transfer Principle asserts that every first order equivalent to x < y if x, y and again the -symbol can statement true over is similarly true over * and vice safely be dropped. versa. This means that every statement is valid for areal The logical connectives (, , , , ) and both if and only if the corresponding formula is valid quantifiers (, ). For all of them the ∗-transform is on ahyperreal , hence the transfer Principle asserts that identical to the inverse image, so that -symbol should every first order statement true over is similarly true be dropped. over and vice versa (Davis, 2009). This means that the The idea of constructing a ∗-transform such that the truth of the statements follows by the transfer principle from first-order of sentence would be true if and only if is the fact that the sentence is true in its standard structure. Now we will introduce some example. true, is called the transfer principle. Example. 4.2.1 The Transfer Principle for First-Order Sentences Theorem. 4.2.1. (Transfer Principle) The following first-order sentence which expressing the Archimedean property of the real numbers, using the A sentence φ in is true if and only if ∗φ in * is true. mathematical logic it can be written as: 44 Jwahir-H-Gdal / American Journal of Applied Sciences 2021, Volume 18: 33.50 DOI: 10.3844/ajassp.2021.33.50 xnxn , (5) is true. Observe that these sentence is not ∗-transform of the L-sentence because of the constant c. We can By applying a transfer principle the equivalent first- replace the constant c by existentially an quantified L variable as L-sentence as follows: order of * -sentence given by: *** xnxn ** (6) b n u(, n b Example.4.2.2 Thus the extensiale transfer yields that: Consider the standard mathematical functions which given by: bnunb ( (x ) cos(-x) = cos x Which prove that the sequence is bounded in . x (x ) cosh x + sinh x = e Proposition. 4.3.2 (x, y ) log xy = log x + log y If f and g are functions of n variables on , then for * is true. Then it can be transferrable to a hyperreal u = (u1,…,un) we have: hence the following facts: (f + g)* (u) = f * (u) + g* (u) and (x ) cos(-x) = cos x (f g)* (u) = f * (u) g* (u), when u dom f * dom g* x (x ) cosh x + sinh x = e f * (u)| = | f (u)|*, when x dom f * (x, y ) log xy = log x + log y Proposition. 4.3.3 Also are true. Since areal number is an ordered field which we If U and V are two sets in n then: expressed it in finite number of L-sentence, by transfer * principle we can conclude that the ∗-transform of these L- = sentence are true hence these explain that is an ordered (AB)* = U * V * field. So, instead of explicitly proving the ordered field (UV)* = U * V * axioms ordered field axioms, we can simply take the ∗- (U')* = (U *)' transforms of list which we mentions in example (4.1.4) of UV then U * V * first-order sentences of, that it is true by the transfer principle. We have thus proven that is a totally ordered which expresses the facts true in * (Goldblatt, 1998). field without ever considering as an ultrapower of , Now we present theorem which also assert that a nor even doing a single ultrafilter calculation. first-order formula in is true if and only if it is true in , so its the transfer principle is direct consequence of. The Existential of Transfer Principle That is Lo's’ theorem which is also sometimes known The existential of transfer principle states that “If there as the fundamental theorem of Ultra products. We give exists a hyperreal number satisfying a certain property then its formal statement below. there exists a real number with this property”. Theorem 4.3.4 (Lo's’ Theorem) This principle can used to conclude that the original 1 2 m sequence must be bounded in areal . For any L formula Q(u1, u2…..um) and any r , r …..r n * 1 m Theorem. 4.3.1 the sentence Q ([r ],…..,[r ]) is true if and only if 1 m Qrr nn,..., is true for almost all n . In other words: *** If the extend hyperal sequence u :, is never takes infinitely large values then the extend *11 mm Q rristrueiff ,...,,..., Q rrF sequence u* is bounded in . Proof The Lo´s’ Theorem include transfer as special case because if Q is a sentence then it has no free variables so Let u: be a sequence, suppose c be an element * that Q(v1,...,vm) is just Q and likewise for Q . Thus * of \ , suppose that the sentence: [Q(r1,...,rm)] is in if Q is true and otherwise, independently of sequences rj. Since Lo's’ Theorem * n ** u( n c in this case simply says, Q is true if Q is true. Which is the transfer principle (Goldblatt, 1998; Staunton, 2013). 45 Jwahir-H-Gdal / American Journal of Applied Sciences 2021, Volume 18: 33.50 DOI: 10.3844/ajassp.2021.33.50 The transfer principle in this formulation expresses The sequence S: has -transform, S*: ** the fact that any classical statement is equivalent to the hence s*(n) = s* for n * non-standard statement obtained by replacing everything n by its ∗-transform except the bound variables in the Theorem. 5.1.1 statement (Keisler, 1976). Areal value sequence sn: n convergence to Theorem 4.3.5 L , iff sn ≃ L, for all unlimited n. For classical statement, Q(u1, u2…..um) with a finite Proof number of constants or free variables v1, ..., vp, then Q(v1, ** Suppose sequence s : n converges to L for N ..., vm) Q v v1 , . . . , m . n * is unlimited we want to show sn ≃ L. Recall that the Proof standard convergence condition implies that > 0 there * * In Los’ theorem take v1 = v1 ….… vm = vm then [Q(v1, is k() so that: * * ..., vm)] , but [Q(v1, ..., vm)] Q(v1, ..., vm). nnksL n One fact is disguised in this formulation of transfer, namely that in the statement to the right the bound is true in . Then by (universal) transfer: variables need not be standard. Now we come to the main result of our study. In the n ** n k S L following section we apply the transfer principle to given a n classical theory of sequences so as to obtain the * nonstandard model equivalents (Hurd and Loeb, 1985). And so for n and N > k() then SLn . Since n is unlimited and k() is limited so sL* for Nonstandard models for Real Valued n Convergence Sequences all unlimited N then we conclude that > 0, sL* 0 for all unlimited n * , hence s ≃ L. In this section we will apply a transfer principle to a n n n give a nonstandard model equivalents for the basic Conversely, suppose sn ≃ L, for an unlimited N theory of convergence sequences. we will illustrate these then nN* if, n>N, hence n is also unlimited therefore results by considering the basic theory of limits, cluster |s -L|<, so, s ≃ L. Let z be an element of * /N, point, Cauchy convergent criterion for real sequences. n n Through this section we will use the fact that a therefore by a hypothesis it is true: hyperreal * is order field which has areal number as nnNsL* a subfield and including unlimited numbers N is a subset n of \ * there for infinitesimals and satisfies a transfer principle (Goldblatt, 2012). Therefore: Nonstandard Model for Convergence theorem. ** znnzsL n Definition. 5.1.1 is true. But a sentence (4.4) formatted existential Let Sn: n , S: be standard sequence of transfer, yields: areal number . Let s(n) = sn, denote the sequence by sn: n which converges to the limit L . Given ε > z n n z sn L 0 in there is a k(ε) so that |sn-L|< ε, for all n>k. Formally standard condition of convergence sequence is true. Which is the desired conclusion that a sequence can expresses by the statement: sn: n is converges. Nonstandard Model for a Limits of a Convergence kxn ks L n . Sequences Remark Theorem. 5.2.1 For ε > 0 the interval (L-ε, L + ε) contains standard Let sn: n and sn: n be standard sequences of real numbers. Then if lim sLn and limtMn in tails of sequence Sn: n i.e., contains the terms n n sn, sn+1, sn+2,……. From some point on it then following properties are holds: 46 Jwahir-H-Gdal / American Journal of Applied Sciences 2021, Volume 18: 33.50 DOI: 10.3844/ajassp.2021.33.50 I. l i m (sn + tn) = L + M then |sn| Areal valued sequence sn:n has at most one limits. Suppose sn: n is bounded then there is a k Proof so that the sentence: Suppose that a sequence s : n converges to L n nNnsM and M then: n * * is true in . By transfer, sn < k for all n , hence |sn | sn = L or, sn ≃ L is finite for all unlimited n. sn = M or, sn ≃ M Conversely, if sn: n is finite for all unlimited n. So by applying transfer the sentence: Therefore we have, M ≃ sn ≃ L, so L ≃ M, hence L = M. ** znsz n Definition. 5.2.1 The standard definition of a limit point L of a real Thus extential transfer (4.3.1) yields that: value sequencesn: n states that, for a given > 0 in and k there is n and n > k, so that |sn-L|<. nsM n Nonstandard Model for Bounded Sequences is true. Which prove that the sequence sn: n is Definition 5.3.1. (The Standard Definitions of bounded in . Bounded Sequences.) Definition 5.3.1 A sequence sn: n is said to be bounded if there Areal value sequences : n is said to be: is M so that the sentence: n Bounded above. If there if there is a real upper nsM n bound to the sequence sn: n if and only if it has no positive unlimited extended terms Theorem. 5.3.1 Bounded Below. If there if there is a real upper bound to the sequence sn: n if and only if it has Areal valued sequence sn: n is abounded in no negative unlimited extended terms if and only if its extended terms are all limited. Nonstandard Model for the Cauchy Sequences Proof Definition. 5.4.1 By universal transfer the sequence sn: n the * * extended sequence |sn| 47 Jwahir-H-Gdal / American Journal of Applied Sciences 2021, Volume 18: 33.50 DOI: 10.3844/ajassp.2021.33.50 m, n . Formally standard Cauchy sequence can characterization is what would be expected, that for expresses by the statement: monotone sequences only one infinite number is needed for convergence. km nm nkss,, mn Nonstandard Model for the a Monotonic Sequences Lemma. 5.4.1 Definition. 5.5.1 Areal value sequence sn: n is said to be standard The sequence sn: n is a Cauchy sequence iff ** monotonic in if it is either: ssnm, for all unlimited n and m. Non decreasing, i.e., if s s Theorem. 5.4.2. )Cauchy Convergence Criterion) 1 2 Non increasing i.e., s1 s2 A real valued sequence sn: n converges in iff it is a Cauchy sequence. Theorem. 5.5.1 Proof A monotonic bounded sequence sn: n converges in if it is: The sequence sn: n is a Cauchy sequence then it is bounded. Suppose m * be unlimited number since I. Bonded above and no decreasing II. Bounded below and no increasing sm is limited then from theorem (5.4.1) hence sm-L ≃ 0, i.e., (sm is close to L in ), because of the fact that all Proof extended terms of a sequence are infinitely closed to each other we conclude that they infinitely close to We shall proof case only I. Let s be extended term L hence sm ≃ L. Hence by theorem (5.2.1) a sequence we show the term s has standard part L and also is a sn: n is converges to L . Conversely if sn: least upper bound of the set {sn: n } in . Then a ** n converges to L then sLs for all unlimited * nm statement s1 s2 b, holds for all n . n, m, then by theorem (5.1.1) and (5.4.1) a sequence sn: So it holds for all n and by particular s1 s2 b, n is a Cauchy sequence. showing that is limited, so indeed has standard part L. Theorem. 5.4.3 Now we show that L is upper bound of the real. Since the sequence sn: n . The sequence sn: n is Cauchy if and only if for Is non-decreasing then by universal transfer we have. n infinitely large values of n-the terms of the sequence are * m sn sm for all n, m . In particular if n then n infinitesimally close (Davis, 2009). , so sn sN ≃ L, giving sn L as both number are real. Proof Now we will show L is the least upper bound in . For if t is any real upper bound of {sn: n }, then by Let s : be a real-valued Cauchy sequence. That * the property of Being Cauchy is first-order defined as: transfer sn t for all n so L ≃ sN t. It follows that L t so L, t (Goldblatt, 2012). Nm nm nNss,, nm Nonstandard Model for a Cluster Point of a Sequences Since s is Cauchy this sentence is true, hence its ∗- transform: Definition. 5.6.1. (Cluster point) Let sn: n be a real valued sequence, a point L **** * Nm nm n N,, s s nm is a cluster point of a sequence sn: n if each interval (L-, L-) contains many terms of a sequence. is also true. It applied to the extended hyperreal We express a standard definition of a cluster point as sequence sn: * . We know that there exists some the following sentences: n kN such that: mnn m s L n n,, mm ,,1 nss*** nm Theorem. 5.6.1 Recall that a sequence is monotone if it is either an A real valued sequence sn: n has a point L increasing or decreasing function. The following is a cluster point iff sN ≃ L for some unlimited N. 48 Jwahir-H-Gdal / American Journal of Applied Sciences 2021, Volume 18: 33.50 DOI: 10.3844/ajassp.2021.33.50 Proof References Suppose the sentence: Abdeljalil, S. (2018). A new approach to nonstandard analysis. Sahand Communications in Mathematical Analysis, 12(1), 195-254. mnnmsL n Andres, D., & Rayo, B. (2015). Introduction to non- standard analysis. * Is hold. Let be appositive infinitesimal and m http://math.uchicago.edu/~may/REU2015/REUPape then by transfer of a sentence: rs/Rayo.pdf Bell, J. L. (2019). Nonstandard Analysis. In The Continuous, the Discrete and the Infinitesimal in **** mnnmsL n Philosophy and Mathematics (pp. 209-213). Springer, Cham. Hence there is some n * with n > m and therefor n Bottazzi, E. (2018). A Transfer Principle for the is unlimited and: Continuations of Real Functions to the Levi-Civita Field. p-Adic Numbers, Ultrametric Analysis and Applications, 10(3), 179-191. sLn 0 Ciurea, G. (2018). A nonstandard approach of helly’ selection principle in complete metric spaces. Proceedings of the Romanian Academy Series A- Thus sn is an extended term infinitely close to. Now conversely suppose there is an unlimited N with Mathematics Physics Technical Sciences Information Science, 19(1), 11-17. sN ≃ L. Let any positive integer and m . Then n Davis, I. (2009). An Introduction to Nonstandard Analysis. > N > m and |sN-L| < this explain that: Internet source, publication date August, 14, 25. de Jong, M. S. (2020). An Introduction to Nonstandard * nnmsL n Analysis (Bachelor's thesis). Duanmu, H. (2018). Applications of Nonstandard Analysis to Markov Processes and Statistical Therefor by extential transfer |s -L| < for some n Decision Theory (Doctoral dissertation). n , for all n > m. Goldblatt, R. (1998). Graduate Texts in Mathematics. Lectures on the Hyperreals: An Introduction to Conclusion Nonstandard Analysis, 188. A nonstandard analysis provides anew Goldblatt, R. (2012). Lectures on the hyperreals: an methodology for mathematics, in particularly for real introduction to nonstandard analysis (Vol. 188). analysis because the availability of infinitesimals Springer Science & Business Media. allows for easier, directly and more inutility natural Goldbring, I. (2014). Lecture notes on nonstandard proofs in a hyperreal system of theorems which is analysis UCLA summer school in logic. holds in areal system. Goldbring, I., & Walsh, S. (2019). An invitation to The theorems we presented can be generalized to nonstandard analysis and its recent applications. transferring a higher order sentences. Notices of the American Mathematical Society, Recommendation for further research work to examine 66(6), 842-851. the nonstandard analysis for diver numbers systems. Hurd, A. E., & Loeb, P. A. (1985). An introduction to nonstandard real analysis. Academic Press. Acknowledgment Katz, M. G., & Polev, L. (2017). From Pythagoreans and Weierstrassians to true infinitesimal calculus. arXiv My thankful goes to referees for their careful and preprint arXiv:1701.05187. helpful remarks. Keisler, H. J. (1976). Foundations of infinitesimal calculus (Vol. 20). Boston: Prindle, Weber & Schmidt. Ethics Keisler, H. J. (2000). Elementary Calculus An This article is original and contains unpublished Infinitesimal Approach. material. The corresponding author confirms that all of Lengyel, E. (1996). Hyperreal structures arising from an the other authors have read and approved the manuscript infinite base logarithm (Doctoral dissertation, and no ethical issues involved. Virginia Tech). 49 Jwahir-H-Gdal / American Journal of Applied Sciences 2021, Volume 18: 33.50 DOI: 10.3844/ajassp.2021.33.50 Mocanu, C. E., Nichita, F. F., & Pasarescu, O. (2020). Staunton, E. (2013). Infinitesimals, Nonstandard Applications of Non-Standard Analysis in Topoi to Analysis and Applications to Finance. Mathematical Neuroscience and Artificial Stroyan, K. D., & Luxemburg, W. A. J. (1977). Intelligence: I. Mathematical Neuroscience. Introduction to the Theory of Infiniteseimals. Ponstein, R. J. (1975). Nonstandard analysis. Academic Press. Robinson, A. (2016). Non-standard analysis. Princeton Sun, Y. (2015). Nonstandard analysis in mathematical University Press. economics. In Nonstandard Analysis for the Working Sanders, S. (2019). A note on non-classical nonstandard Mathematician (pp. 349-399). Springer, Dordrecht. arithmetic. Annals of Pure and Applied Logic, 170(4), 427-445. 50