GEORG CANTOR (1845-1918) AND THE "DEGREES OF THE INFINITE"
0)
Since Zeno's times (5th century before Christ), philosophers have been discussing about the "infinite", both in Theology and in Mathematics, but no one before 1872 had been able to explain exactly what he was speaking or writing about (Carl Boyer, "A History of Mathematics").
These lessons deal with some of the subjects that Cantor studied.
1)
What is a "set"?
A "set" is a "collection", a "group" of objects, which are called "the elements" (or: "the members") of the set.
·
the element x “belongs to” the set A.
A "subset" of a given set A, is a "part" of A;
it is a set such
that, if ,
then
.
·
B is a subset of A, B is included in A
Let A be the
following set:
Then the
subsets of A are:
Among the subsets of a set A, there is also A itself.
We say that A is an "improper" subset of A.
The other subsets of A (the ones that do not coincide with A) are called "the proper subsets" of A.
(Remark: some textbooks include the empty set among the improper subsets).
2)
There are some numbers that cannot be expressed by a fraction, that is, by a quotient of two integers. For example, the following statement can be proved:
THEOREM. - The number cannot be expressed in the form of a
fraction;
in other words, there isn't any fraction that, if raised to square, produces 2 as a result.
The existence of numbers that cannot be expressed by a fraction can also be proved by the following reflections.
Every time we take a fraction and change it into a decimal number, by dividing, we'll certainly find:
a) either a finite decimal number
b) or a "repeater".
I mean that we'll never find a decimal number without the "period" (= the group of digits that repeats itself endlessly).
The reason of this lies in the fact that, in a division, the "remainder" is always smaller than the "divisor". So, when we consider the fraction a/b and take it to decimal form, in every step of the division algorithm we'll find one of the following remainders:
0, 1, 2, .. , b -1.
The number of possible remainders is limited: so, it is sure that, sooner or later,
a) either we'll find 0 as a remainder, and therefore stop the division
b) or, otherwise, we'll meet again a remainder that had already occurred before
Now, the following decimal number: x =0,1010010001000010000010000001…
is NOT a repeater.
So, no fraction can be equivalent to x.
In general, if a decimal number is NOT a finite decimal or a repeater, it CANNOT be reduced to a fraction (=quotient of integers). We say that it is an "irrational" number.
We denote the set of all numbers, rational or irrational, with the symbol R .
3)
We can represent real numbers by means of a straight line (the so-called "number line").
An accurate treatment of this subject leads to assume that:
to every real number x, it corresponds a point P of the line, and conversely, every point Q on the line is the corresponding of a certain real number x'.
We say that, between the set of real numbers and the set of the points of the number line, there is a "one-to-one correspondence".
4)
In general:
DEFINITION. - We say that there is a "one-to-one correspondence" between two sets A, B,
when it happens that to every element of A, it corresponds one and only one
element of B, and conversely.
Instead of saying "one-to-one correspondence" we can also say "bijection".
5)
DEFINITION. -We say that two sets A, B are "equipotent" if there is a bijection between A and B. |
6)
|
THEOREM ("Galileo's Paradox"): The set N* = { 1, 2, 3,...} of positive integers is equipotent with the set E={ 2, 4, 6, ...} of even integers. (Yet, E is just a "proper subset" of N*!!!)
|
Proof.:
7)
The foregoing remarks suggest an idea to formulate a rigorous definition of "infinite set".
|
DEFINITION. - We say that a set is "infinite" if it is equipotent with a "proper subset" of itself; otherwise, we say that the set is "finite". |
8)
N = { 0, 1, 2, 3, ... } is equipotent with N* = { 1, 2, 3, ... }:
Therefore, N is also equipotent with E = { 2, 4, 6, ...}.
It can easily be proved that N, N*, E are equipotent with the set F = {1, 3, 5, ...} of odd integers.
All the previously mentioned sets are also equipotent with Z = { ...,-3, -2, -1, 0, 1, 2, 3, ... }
(Z is called "the set of relative integers"):
9)
|
DEFINITION. - We call "cardinal number" the abstract entity which is common to all the sets that are equipotent with a given set. |
|
3 three trois
tre
|
A cardinal number is said to be "infinite" or "finite" depending on whether it is represented by an "infinite set" of by a "finite set" (see def. 7).
It can easily be proved that this definition is "correct", since it does not depend on the particular set which is used to "represent" the cardinal number we want to consider.
The cardinal number which is represented by the set of the fingers of a hand is usually indicated by the symbol " 5 ";
The cardinal number which is represented by an empty set is usually indicated by the symbol " 0 ";
...
The cardinal number which is represented by the set N*
(or N, or E, or F, or Z … ) is usually
denoted as "aleph zero" ;
"aleph" is the first letter of the Hebrew
alphabet .
Remark: Some textbooks use, in place of “aleph0”, the symbol d (from the English word "denumerable")
|
|
“aleph zero”
the cardinality of denumerable sets la “cardinalità del numerabile” (si dice anche “la potenza del numerabile”); insomma, l’entità astratta che è comune a tutti gli insiemi che possono essere messi in corrispondenza biunivoca con l’insieme degli interi positivi.
|
|
So far, we have learned that:
card { card {Paperino} = 1 card {Paperino, Paperone} = 2 card {Qui, Quo, Qua} = 3 … card (N) = card (N*) = card (E) = card(F) = card(Z) = aleph0
|
10)
Let's consider two cardinal numbers a, b and two sets A, B such that card (A) = a, card (B) = b.
We say that a<b if A is equipotent with a subset of B, but B isn't equipotent with any subset of A.
Given two cardinal numbers a, b, we call "the sum" of a and b (a+b), the cardinal number of the set A U B, provided that
card (A) = a, card (B) = b,
and A, B are disjoint sets.
Given two cardinal numbers a, b, we call "the product" of a and b (aּb, ab )
the cardinal number of the set A X B (the "Cartesian product" of A and B, namely the set whose
elements are the ordered pairs (x, y), where x is an element of A and y is an element of B),
provided that
card (A) = a, card (B) = b
It can be proved that the foregoing definitions are "correct", since they don't depend on the particular sets which are used to "represent" the cardinal numbers.