Natural Integer and Rational numbers The natural numbers

Natural, Integer and Rational numbers • The natural numbers are best given by the axioms of Giuseppe Peano (1858 - 1932) given in 1889. 1. 2. 3. 4. 5. • • • There is a natural number 0. Every natural number a has a successor, denoted by a + 1. There is no natural number whose successor is 0. Distinct natural numbers have distinct successors: if a ≠ b, then a + 1 ≠ b + 1. If a property is possessed by 0 and also by the successor of every natural number it is possessed by, then it is possessed by all natural numbers. Richard Dedekind (1831 -1916) proved that any model of above axioms is isomorphic to the natural numbers. In particular one obtains a particular model N from the Zermelo-Fraenkel axioms. The set of natural number carries two operations addition and mulitplication. Adding the inverse operation of addition namely subtraction, one obtains the integers Z. Adding the inverse operation to multiplication to Z one arrives at the rational numbers Q. As Kronecker puts it: Die ganze Zahl schuf der liebe Gott, alles Übrige ist Menschenwerk. (God made the integers, all else is the work of man. )

• • The real numbers Real numbers are commonly associated to the points on a line. Another way to think of them is as the limit of rational numbers or nested intervals. Both observations lead to definitions of the reals starting from the rationals. The important property here is that the sequence of nested intervals need not converge in Q. Axioms for the real numbers: 1. The set R is a field, i. e. , addition, subtraction, multiplication and division are defined and have the usual properties. 2. The field R is ordered, i. e. , there is a linear order ≥ such that, for all real numbers x, y and z: 3. if x ≥ y then x + z ≥ y + z; 4. if x ≥ 0 and y ≥ 0 then xy ≥ 0. 5. The order is Dedekind-complete, i. e. , every non-empty subset S of R with an upper bound in R has a least upper bound (also called supremum) in R. • • Dedekind defined the real numbers via so-called cuts A Dedekind cut is a subset a of the rational numbers Q that satisfies these properties: 1. a is not empty. 2. Q a is not empty. 3. a contains no greatest element 4. For x, y in Q if x in a and y < x , then y is in a as well. • Real numbers can then be defined as the set of Dedekind cuts.

The real numbers as Cauchy sequences • • • Cantor defined the real numbers through Cauchy sequences. A Cauchy sequence of rationals is a sequence rn in Q indexed by the natural numbers N, such that for any rational e there is a natural number N s. t. |rm-rn|<e for all m, n >N Cauchy sequences (xn) and (yn) can be added, multiplied and compared as follows: – (xn) + (yn) = (xn + yn) – (xn) × (yn) = (xn × yn) – (xn) ≥ (yn) if and only if for every ε > 0, there exists an integer N such that xn ≥ yn - ε for all n > N. • • Two Cauchy sequences (xn) and (yn) are called equivalent if such that for any rational e there is a natural number N s. t. |xn-yn|<e for all n >N The real numbers are now defined to be equivalence classes of rational Cauchy sequences. The real numbers are then complete, i. e. all Cauchy sequences of real numbers converge. Lastly one can use the decimal expansion, which essentially selects a representative for each class, with the proviso that some reals have two decimal expansions which are equivalent, e. g. 1. 0000 … = 0. 99999….

Hyperreal numbers • In order to make sense of infinitesimals one should think of them as sequences of real numbers (xn). • we can also add and multiply sequences: (a 0, a 1, a 2, . . . ) + (b 0, b 1, b 2, . . . ) = (a 0 + b 0, a 1 + b 1, a 2 + b 2, . . . ) and analogously for multiplication. • Hyperreals are defined as equivalence classes of sequences of reals. • Two sequences will be equivalent is there difference has infinitely many zeros. • It turns out, this is not enough, though to get multiplication and an order. In technical detail the slightly larger equivalence is given by an so-called ultrafilter U on the natural numbers which does not contain any finite sets. Such an U exists by the axiom of choice. One can think of U as singling out those sets of indices that "matter“ when comparing two sets: We write (a 0, a 1, a 2, . . . ) ≤ (b 0, b 1, b 2, . . . ) if and only if the set of natural numbers { n : an ≤ bn } is in U. This is a total preorder and it turns into a total order if we agree not to distinguish between two sequences a and b if a≤b and b≤a. • *R is then given by sequences in R modulo the above equivalence. • *R has addition, subtraction multiplication, and divison. • The real numbers are the subset given by the constant sequences (r, r, r. . ) This construction is unique if the continuum hypothesis holds

Inifitesimals • *R contains the finite numbers, F={*r in *R| |*r|<r for some r in R} • An infinite number is e. g. given by the sequence (1, 2, 3, 4, …. ) • *R contains the infinitesimal numbers. Infinitesimals = {*r in *R| |*r|<r for all r in R} • An infinitesimal is given by the sequence (1, 1/2, 1/3, …) • Each finite hyperreal *r has a unique standard part st(*r) which is defined to be the unique number s. t. *r-st(*r) is infinitesimal. • Given a function f: R→R, it has a unique extension *f to *R given by *f (a 0, a 1, a 2, . . . ) : =(f(a 0), f(a 1), f(a 2), . . . ) • Also one writes x≈y if x-y is infinitesimal • In this notation f is continuous if *f(x+h)) ≈f(x) for all infinitesimals h. • Fix an infinitesimal dx then – df(x): = *f(x+dx)-f(x) – df(x)/dx= (*f(x+dx)-f(x))/dx is always defined in *R, if it is finite for all reals x and all infinitesimals dx then f is differentiable and df(x)/dx≈f’(x).