Александра Владимирова Цветкова, ф. № 43605
Александра Владимирова Цветкова, ф. № 43605
ФМИ, Информатика, 5 гр., 1 курс
Курсова работа по английски език, 2003/2004
Georg Ferdinand Ludwig Philipp Cantor
http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Cantor.html
At Halle the direction of Cantor's research turned away from number theory
and towards analysis. This was due to Heine, one of his senior colleagues at Halle,
who challenged Cantor to prove the open problem on the uniqueness of representation
of a function as a trigonometric series. This was a difficult problem, which had been
unsuccessfully attacked by many mathematicians, including Heine himself as well as
Dirichlet, Lipschitz and Riemann. Cantor solved the problem proving uniqueness of
the representation by April 1870. He published further papers between 1870 and 1872
dealing with trigonometric series and these all show the influence of Weierstrass's
teaching.
Cantor was promoted to Extraordinary Professor at Halle in 1872 and in that
year he began a friendship with Dedekind who he had met while on holiday in
Switzerland. Cantor published a paper on trigonometric series in 1872 in which he
defined irrational numbers in terms of convergent sequences of rational numbers.
Dedekind published his definition of the real numbers by "Dedekind cuts" also in
1872 and in this paper Dedekind refers to Cantor's 1872 paper which Cantor had sent
him.
In 1873 Cantor proved the rational numbers countable, i.e. they may be placed
in one-one correspondence with the natural numbers. He also showed that the
algebraic numbers, i.e. the numbers that are roots of polynomial equations with
integer coefficients, were countable. However his attempts to decide whether the real
numbers were countable proved harder. He had proved that the real numbers were not
countable by December 1873 and published this in a paper in 1874. It is in this paper
that the idea of a one-one correspondence appears for the first time, but it is only
implicit in this work.
A transcendental number is an irrational number that is not a root of any
polynomial equation with integer coefficients. Liouville established in 1851 that
transcendental numbers exist. Twenty years later, in this 1874 work, Cantor showed
that in a certain sense 'almost all' numbers are transcendental by proving that the real
numbers were not countable while he had proved that the algebraic numbers were
countable.
Cantor pressed forward, exchanging letters throughout with Dedekind. The
next question he asked himself, in January 1874, was whether the unit square could be
mapped into a line of unit length with a 1-1 correspondence of points on each. In a
letter to Dedekind dated 5 January 1874 he wrote [1]:
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Александра Владимирова Цветкова, ф. № 43605
Can a surface (say a square that includes the boundary) be uniquely
referred to a line (say a straight line segment that includes the end
points) so that for every point on the surface there is a corresponding
point of the line and, conversely, for every point of the line there is a
corresponding point of the surface? I think that answering this
question would be no easy job, despite the fact that the answer seems
so clearly to be "no" that proof appears almost unnecessary.
The year 1874 was an important one in Cantor's personal life. He became
engaged to Vally Guttmann, a friend of his sister, in the spring of that year. They
married on 9 August 1874 and spent their honeymoon in Interlaken in Switzerland
where Cantor spent much time in mathematical discussions with Dedekind.
Cantor continued to correspond with Dedekind, sharing his ideas and seeking
Dedekind's opinions, and he wrote to Dedekind in 1877 proving that there was a 1-1
correspondence of points on the interval [0, 1] and points in p-dimensional space.
Cantor was surprised at his own discovery and wrote:
I see it, but I don't believe it!
Of course this had implications for geometry and the notion of dimension of a
space. A major paper on dimension, which Cantor submitted to Crelle’s Journal in
1877 was treated with suspicion by Kronecker, and only published after Dedekind,
intervened on Cantor's behalf. Cantor greatly resented Kronecker's opposition to his
work and never submitted any further papers to Crelle's Journal.
The paper on dimension, which appeared in Crelle’s Journal in 1878, makes
the concepts of polynomial equations precise. The paper discusses denumerable sets,
i.e. those which are in 1-1 correspondence with the natural numbers. It studies sets of
equal power, i.e. those sets, which are in 1-1 correspondence with each other. Cantor
also discussed the concept of dimension and stressed the fact that his correspondence
between the interval [0, 1] and the unit square was not a continuous map.
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Александра Владимирова Цветкова, ф. № 43605
Glossary of words and names in their mathematical meaning
(Merriam-Webster's Collegiate Dictionary)
Cantor, Georg Ferdinand Ludwig Philipp – (1845 - 1918) German
mathematician who founded set theory and introduced the mathematically meaningful
concept of transfinite numbers, indefinitely large but distinct from one another.
Heine, Eduard – (1821-1881) German mathematician who made valuable
contribution to analysis;
Dirichlet, Peter Gustav Lejeune – (1805 - 1859) German mathematician who
made valuable contributions to number theory, analysis and mechanics.
Lipschitz, Rudolf - (1832-1903) German mathematician who worked on the
number theory and differential geometry;
Riemann, (Georg Friedrich) Bernhard – (1826 - 1866) German mathematician
whose work widely influenced geometry and analysis.
Weierstrass, Karl (Theodor Wilhelm) – (1815 – 1897) German
mathematician, one of the founders of the modern theory of functions.
Dedekind, (Julius Wilhelm) Richard – (1831 – 1916) German mathematician
who developed a major redefinition of irrational numbers in terms of arithmetic
concept.
Liouville, Joseph – (1809 - 1882) French mathematician known for his work
in analysis, the theory of numbers, and differential geometry, and particularly for his
discovery of transcendental number –i.e., numbers that are not the roots of algebraic
equations having rational coefficients.
Crelle, August Leopold – (1780 - 1855) German mathematician and engineer
who advanced the work and careers of many young mathematicians of his day and
founded the Journal für die reine und angewandte Mathematik (“Journal for Pure and
Applied Mathematics”), now known as Crelle’s Journal.
Kronecker, Leopold - (1823-1891) German mathematician who worked on
number theory;
"Dedekind cuts” – Dedekind’s method, now called the Dedekind cut,
consisted in separating all the real numbers in a series into two parts such that each
real number in one part is less than every real number in the other. Such a cut, which
corresponds to a given value, defines an irrational number if no largest or no smallest
is present in either part; whereas a rational is defined as a cut in which one part
contains a smallest or a largest. Dedekind would therefore define the square root of 2
as the unique number dividing the continuum into two collections of numbers such
that all the members of one collection are greater than those of the other, or that cut,
or division, separating a series of numbers into two parts such that one collection
contains all the numbers whose squares are larger than 2 and the other contains all the
numbers whose squares are less than 2.
Number theory - a branch of mathematics that grew out of human curiosity
concerning properties of the positive integers 1, 2, 3, 4, 5, ... also called whole
numbers, natural numbers, or counting numbers;
Irrational number - a number that can be expressed as an infinite decimal
with no set of consecutive digits repeating itself indefinitely and that cannot be
expressed as the quotient of two integers;
Rational number - an integer or the quotient of an integer divided by a
nonzero integer;
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Александра Владимирова Цветкова, ф. № 43605
Real number - one of the numbers that have no imaginary parts and comprise
the rationals and the irrationals;
Natural number - the number 1 or any number (as 3, 12, 432) obtained by
adding 1 to it one or more times - a positive integer;
Algebraic number - a root of an algebraic equation with rational coefficients;
Transcendental number - incapable of being the root of an algebraic
equation with rational coefficients (Pi is a transcendental number);
Coefficient - any of the factors of a product considered in relation to a specific
factor; esp: a constant factor of a term as distinguished from a variable;
Denumerable - capable of being put into one-to-one correspondence with the
positive integers;
Set - a number of things of the same kind that belong or are used together;
Analysis - a branch of mathematics concerned mainly with functions and
limits;
Geometry - a branch of mathematics that deals with the measurement,
properties, and relationships of points, lines, angles, surfaces, and solids;
Function - mathematical correspondence that assigns exactly one element of
one set to each element of the same or another set;
Trigonometric - of, relating to, or being in accordance with trigonometry;
Series - the indicated sum of a usu. infinite sequence of numbers;
Convergent (a ~ sequence) - characterized by having the nth term or the sum
of the first n terms approach a finite limit;
Sequence - a set of elements ordered so that they can be labeled with the
positive integers;
Polynomial (~ equations) - relating to, composed of, or expressed as one or
more polynomials (a mathematical expression of one or more algebraic terms each of
which consists of a constant multiplied by one or more variables raised to a
nonnegative integral power);
Equation - a usu. formal statement of the equality or equivalence of
mathematical or logical expressions;
One-one - pairing each element of a set uniquely with an element of another
set;
Correspondence - a relation between sets in which each member of one set is
associated with one or more members of the other;
Root - a quantity taken an indicated number of times as an equal factor (2 is a
fourth ~ of 16);
Unit - a single quantity regarded as a whole in calculation;
Surface - a plane or curved two-dimensional locus of points (as the boundary
of a three-dimensional region), for example – a plane surface, surface of a sphere;
Square – 1. a product of a number multiplied by itself; 2.a rectangle with all
four sides equal;
Boundary - something (as a line, point, or plane) that indicates or fixes a limit
or extent;
Segment – 1. a portion cut off from a geometric figure by one or more points,
lines, or planes; 2. a part of a circular area bounded by a chord and an arc of that
circle or so much of the area as is cut off by the chord; 3. a part of a sphere cut off by
a plane or included between two parallel planes; 4. a finite part of a line between two
points in the line;
Interval - a set of real numbers between two numbers either including or
excluding one or both of them;
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Александра Владимирова Цветкова, ф. № 43605
P-dimensional - relating to or having p dimensions; esp: consisting of or
relating to elements requiring p coordinates to determine them;
Space - a set of mathematical elements and esp. of abstractions of all the
points on a line, in a plane, or in physical space; esp: a set of mathematical entities
with a set of axioms of geometric character;
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Александра Владимирова Цветкова, ф. № 43605
Exercise:
1.
a)
b)
c)
d)
2.
a)
b)
c)
d)
3.
a)
b)
c)
d)
4.
a)
b)
c)
d)
5.
a)
b)
c)
d)
6.
a)
b)
c)
d)
7.
a)
b)
c)
d)
When was realized the first contact between Cantor and Dedekind?
1870;
1874;
1873;
1872;
Rational numbers may be placed in one-one correspondence with :
the natural numbers.
the real numbers.
the transcendental numbers .
the algebraic numbers.
Who established that transcendental numbers exist?
Dedekind;
Leouville;
Cantor;
Kronecker.
Which of the following was not a part of Cantor’s works?
trigonometric series;
set theory;
complex numbers;
polynomial equations.
Which of the following Cantor hadn’t worked with?
Heine;
Dedekind;
Louville;
Dirichlet.
What is an algebraic number?
a root of an algebraic equation with rational coefficients;
a number that can be expressed as an infinite decimal with no set of
consecutive digits repeating itself indefinitely and that cannot be expressed
as the quotient of two integers;
a root of an algebraic equation with natural coefficients;
an irrational number that is not a root of any polynomial equation with
integer coefficients.
Which of Cantor’s works was published in Crelle’s Journal?
1-1 correspondence of points on the interval [0, 1] and points in pdimensional space;
trigonometric series defining irrational numbers in terms of convergent
sequences of rational numbers.;
polynomial equations;
transcendental numbers.
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