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of the Real Number System |
The purpose of this material is to present an algebraic structure that has the properties of infinitesimals and an algebraic structure that is isomorphic to the real numbers. The real number system is an ordered mathematical field.* After the presentation the result will be related to the subtle and profound nonstandard analysis of Abraham Robinson.
It is shown elsewhere that there is a need for numbers beyond the reals to describe orders. The Greek mathematician Archimedes long ago demonstrated the power of the concept of infinitesimals. For example, he derived the formula for the area enclosed with in a circle, by considering that area as composed of infinitesimal triangles with an apex at the center and a base on the circle.
The area of such a triangle is one half of the height of the triangle, in this case the radius r of the circle, times the length of the base. The total length of the bases of all the triangles is just the circumference of the circle, 2πr. Thus the area enclosed by a circle of radius r is ½(r)(2πr), which is equal to πr². Likewise the volume of a ball enclosed within a sphere of radius r is sum of all the infinitesimal pyramids with their apices at the center and their bases on the sphere. The volume of a pyramid is one third of the product of its height times the area of its base. The height of each pyramid is the sphere radius r. The sum of the area of the bases is just the area of the sphere 4πr². Thus the volume of the ball is (1/3)r(4πr²) which is equal to (4/3)πr³.
At this point let it be noted that an algebraic structure is usually of the form {S, f, g, h} where S is a set of elements and f, g and h are functions defining the operations such as addition, multiplication and order relation. There is a tendency to focus attention on the elements of the set, but these are irrelevant; it is the functions that carry the content.
When Isaac Newton and Gottfried Wilhelm Leibniz first formulated differential calculus they effectively made use of the concept of an infinitesimal, which they referred to as an infinitely small number, whatever that was supposed to mean. For example, Leibniz, who introduced the d notation for differentials, said in deriving d(xy)=xdy+ydx that this follows from
and
the omission of the quantity dxdy, which is infinitely small in comparison with the rest, for it is supposed that dx and dy are infinitely small.
In effect he was saying that the although infinitesimals are not zero the product of infinitesimals is zero.
Thus the term infinitesimal can be defined in several ways. One approach is to define an infinitesimal as an entity such that its square is zero. This approach is explored in The Calculus of Infinitesimals. The notion that is used here is that a positive infinitesimal is something that is smaller than any positive real number.
A non-terminating sequence of any set is just a function from the natural numbers (positive integers) to that set. Consider the set of all nonterminating sequences of real numbers that have limits. A sequence S={s_{1}, s_{2}, …} has a limit L if for any given δ>0 there exists and an integer N such that for all integers n≥N |s_{n}−L|<δ.
Addition and multiplication for such sequences are easily defined. If R={{r_{1}, r_{2}, …} and S={s_{1}, s_{2}, …} then
Associativity follows from the associativity in the real number system.
Let the ordering of the sequences be that R<S if there exists an N such that for all n>N r_{n}<s_{n}. Since only sequences which have a limit are being considered this establishes an order or the equality of any two such sequences.
The real number system can be imbedded in the set S by the mapping
Let {k, k, …} be denoted as {k}.
Now consider the sequence
The limit of this sequence is 0.
For any real k greater than zero
Thus ε_{0} is an infinitesimal. Likewise any sequence ε such that its limit is zero is an infinitesimal.
It can easily be shown that
Let H be the set of the sum of a {k} for k a real number and an infinitesimal ε. Since {0} is an infinitesimal the set H includes the the system of constant sequences {k} for k a real numbers which is isomorphic to the real numbers. Since {0} is equivalent to the real number zero the set H also includes the infinitersimals.
The set H with the operations of addition and multiplication constitute an extention of the real number system. The element {0} is the additive identity and {1} is the multiplicative identity. Any element h of H has an additive inverse; i.e.,
Some elements of H have multiplicative inverses, but those with any components which is zero do not. Thus H is not a perfect (complete) extention of the real number system. However consider for each h an h^{(-1)} such that h^{(-1)}_{n} = 1/h_{n} if h_{n}≠0 and h^{(-1)}_{n}=ℵ if h_{n}=0.
For now consider ℵ to be some arbitrary large real number such as a googol (10^{100}) or a googleplex (10^{googol}. This h^{(-1)} can be called its pseudo-inverse of h, which may or may not be the actual multiplicative inverse of h.
The product of any h and its pseudo-inverse is a sequence that consists of 1's and 0's. It would be nice to be able to say that one of these sequences is almost everywhere equal to the multiplicative identity {1}, but such is not the case. There are sequences which terminate in an infinite sequence of zeroes that, of course, have a limit of zero.
Consider a sequence R={r_{n}; n=1, 2, …} such that for any real k there is an N such that for all n>N r_{n}>k. Such sequences are often said to have a limit of infinity. Since R>{k} for any real k R can be said to be an infinite number. Because of the plethera of connotations of infinite it is better to call such numbers unlimited. It follows easily that
In order for each unlimited number to have an additive inverse it is necessary to include the notion of negative infinities. Generally the difference of unlimited numbers is indeterminant as to type. An unlimited number which has no zero component has a mulitplicative inverse which is an infinitesimal.
Algebraic systems can be formulated axiomatically, in which case there is no question of existence but only the matter of whether the axioms are logically consistent. Alternatively algebraic systems such as the real numbers can be built up out of known elements, as is the case where Cauchy derived the real numbers from converent sequences of rational numbers or Dedekind from partitions of the set of rational numbers. Such derivations can be called models of the real numbers.
The notion that something like the infinitesimals could be congered up by simply defining them as quantities that are less than any real positive number offends mathematical sensibilities. Here the question of existence is paramount.
What Abraham Robinson did was prove that for a set of properly formulated propositions there exists a model such that those propositions are true.
The more exact statement of Robinson's Compactness Theorem (from Goldblatt) is
If a set Σ of statements (of an appropriate kind) has the property that each finite subset Σ' of Σ has a model (a structure in which all members of Σ' are true), then there must be a single structure which is a model of Σ itself.
For example, let Σ consist of all appropriate statements concerning the natural numbers together with the infinitely many statements
These say, there exists a quantity greater than 1; there exists a quantity greater than 2 and so forth.
Then by the Compactness Theorem there is a model I which has the properties of the natural numbers plus has a quantity that is greater than all of the natural numbers, a quantity that could be called infinite,. Furthermore this model I satisfies Robinson's Transfer Principle; i.e., Any appropriately formulated statement is true of I if and only if it is true of the natural number system.
Abraham Robinson then went on to formulate an extension of the real number system which is called the hyperreals, which include infinitesimals. A hyperreal number consists of a real number and a halo of infinitesimals. There are also infinities of the nature presented previously as unlimited numbers. The multiplicative inverse of an infinity is an infiniessimal and vice versa.
(To be continued.)
*Being a field means that there are two operations defined, say addition and multiplication, such the sum and product are defined for every element. These operation are associative and there is an additive identity (zero) and there is a multiplicative identity (unity). For each element there is an additive inverve (negative) and for all elements except the additive identity (zero) there is a multiplicative identity (reciprocal). Being ordered means that for any two elements their order is defined.
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