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Cauchy sequences which they called
fundamental sequences
(1872). After
reading Heine's and Cantor's treatments, Dedekind decided to publish
work that he had done in 1858, when he acknowledged the
unprovability of the convergence of monotonic bounded sequences from
a geometric standpoint. Dedekind described points on the number line
(whether rational or irrational) as producing a cut in the rational
numbers, separating those above the cut from those below. This
description is similar to that in Euclid, Book V, but Dedekind's new
definition of numbers as cuts in the rationals led to formal proofs of
their algebraic properties and of completeness, in that every cut of this
new system is actually a cut of the rationals.
Independently of the German tradition, C. Me´ ray, a student of
Cauchy, had proved that the convergence of bounded monotonic
sequences implied the convergence of Cauchy sequences by considering
their eventual upper bounds and eventual lower bounds respectively. In
the same paper (1869) Me´ ray also defined irrational numbers as
fictitious limits of rational Cauchy sequences.
Defining a real number as a pair of subsets of the rationals (as
Dedekind) or as a family of Cauchy sequences (as Me´ ray and Cantor)
seemed to divorce numbers which originate in counting and
measurement from their roots. So in 1882, P. du Bois-Reymond
proposed as a postulate that every decimal number (whether infinite or
not) corresponded to a unique point on an orientable line, and it is this
proposal which has inspired the choice of
completeness principle
in this
topic. In thesamespirit, Ascoli proposed as a postulatein 1895 that a
set of nested closed intervals with length tending to zero had a unique
common point.
All of this work presumed that the rational numbers were
well-founded and that it was only irrationals that were not precisely
defined. A modern, axiomatic, approach avoids any notion that some
numbers are more 'real' than others, but the fact that the real numbers
can be constructed from the rational numbers (whether by Dedekind
cuts, families of Cauchy sequences, infinite decimals or nested intervals)
establishes that there is nothing inherently contradictory in introducing
a principle (or axiom) of completeness.
Galileo noted in his
Discourses on Two New Sciences
(1638) that the
mapping
n
n
matched the set N, one-to-one, with a proper subset of
N
x
matched the set
x
0
x
2
, one-to-one, with the set
x
0
x
1
, another
one-to-one matching of a set with a proper subset. Either one of these
mappings provides a counter-example to Euclid's axiom: the whole is
always greater than the part. It was Bolzano (1851) who first recognised
that this was always a possibility with infinitests.
. Healso noted that themapping
x
