Natural, Integer and Rational numbers

1
Natural, Integer and Rational numbers

• The natural numbers are best given by the axioms
  of Giuseppe Peano (1858 - 1932) given in 1889.
• 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.)
2
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
• The set R is a field, i.e., addition,
  subtraction, multiplication and division are
  defined and have the usual properties.
• The field R is ordered, i.e., there is a linear
  order such that, for all real numbers x, y and
  z
• if x y then x z y z
• if x 0 and y 0 then xy 0.
• 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
• a is not empty.
• Q \ a is not empty.
• a contains no greatest element
• For x,y in Q if x in a and y lt x , then y is in
  a as well.
• Real numbers can then be defined as the set of
  Dedekind cuts.

3
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-rnlte for all m, n gtN
• 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 e gt 0, there
  exists an integer N such that xn yn - e for all
  n gt 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-ynlte for all n gtN
• 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 .

4
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 (a0, a1,
  a2, ...) (b0, b1, b2, ...) (a0 b0, a1 b1,
  a2 b2, ...) 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 (a0,
  a1, a2, ...) (b0, b1, b2, ...) 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 ab and ba.
• 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
5
Inifitesimals

• R contains the finite numbers, Fr in R
  rltr 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 rltr for all r in
  R
• An infinitesimal is given by the sequence
  (1,1/2,1/2,1/3, )
• 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 (a0, a1, a2, ...)
  (f(a0),f(a1),f(a2), ...)
• Also one writes xy if x-y is infinitesimal
• In this notation f is continuous if f(xh))
  f(x) for all infinitesimals h.
• Fix an infinitesimal dx then
• df(x) f(xdx)-f(x)
• df(x)/dx (f(xdx)-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)/dxf(x).