16)

So we have proved that Z, Q_a Q are all equipotent with N* = { 1, 2, 3, ... }.

Now, it would be perfectly natural to believe that any infinite set is equipotent with N*.

Most surprisingly, it is not so:

there exist sets whose cardinal number is greater than aleph₀!

In fact, George Cantor proved the following fundamental

THEOREM. -

There can be no one-to-one correspondence between the set R of real numbers and the set

N* = { 1, 2, 3, ... }

In other words:

· R is not "countable";

· the cardinal number of R is greater than the cardinal number aleph₀ of the set N*;

· the "degree of infinity" of the set R is greater than the "degree of infinity" of N*;

· R and N* are both infinite, but R is "more infinite" than N* (and so, also than Q)

Proof:

Ab absurdo.

Let’s suppose that there exists a one-to-one correspondence between R and N*; this implies that the elements of R can be arranged in a list

FIRST

COLUMN S E C O N D C O L U M N

In the first column we find all the elements of N*, namely: all the positive integers,

while in the second column we find all the REAL numbers (in their decimal expansion).

But we will soon demonstrate that such a table cannot exist!

Let’s consider the number which is defined as follows:

The integer part of is 2 ( )

The n-th decimal digit of is:

1, if the n-th decimal digit of the real number in the second column of the table is different from 1;

0, if the n-th decimal digit of the real number in the second column of the table is 1

By the way it is defined, cannot occur in the list!

So, the second column does not contain all the elements of R, and the correspondence between N* and R which is represented in the table is not a bijection.

This shows that, if we assume that there is a one-to-one correspondence between R and N*, we are led to absurd conclusions.

17) It can be proved that R is equipotent with any open interval (a, b); more generally, with any interval , no matter if this interval is open, closed, semi-closed, limited or not.

18)

The cardinal number of R

(or of an interval of R)

is indicated by the symbol “c”

and is called "the cardinality of the continuum"

Questo termine “cardinalità del continuo” (o “potenza del continuo”) viene usato per ricordare una delle proprietà fondamentali di R: la "continuità", che appare in qualche modo intimamente connessa col fatto che il grado di infinito di R sia maggiore del grado di infinito del suo sottoinsieme Q (che è denso, ma discontinuo).

So far, we have learned that:

card { } = 0

card {Paperino} = 1

card {Paperino, Paperone} = 2

card {Qui, Quo, Qua} = 3

…

card (N) = card (N*) = card (Q_a) = card (Q) = aleph₀

card (R) = c

N, N*, Q are “denumerable” sets, whilst R is not denumerable.

19) DEFINITION. -

Let S be a given set.

We call "the set of the parts of S" (and we denote it by the symbol P(S) ),

the set whose elements are all the subsets of S.

Let S be the following set:

Then

20)

THEOREM. -

Let A be a set.

Then, the cardinal number of P(A) is greater then the cardinal number of A.

Proof:

We want to show that there isn’t any one-to-one correspondence between A and P(A).

Ab absurdo.

Let’s assume the existence of such a bijection (we’ll indicate it by ):

If we take an element of the set , then is an element of , namely a subset of :

, .

So, there can be two possibilities: or .

Let’s define the set as follows:

= the set of the elements of A, which do not belong to their corresponding set .

Now, let’s consider the “inverse image” of B, that is to say:

let’s consider the element , such that

There are two possibilities:

But neither of them can be true! In fact:

· It’s impossible that , since is, by definition, the set of the elements of A that do NOT belong to their f-image;

· It’s impossible that , since, if it were so, should belong to B, by definition of B.

This theorem tells us that the "degrees of the infinite"

are, in their turn, infinite!!!

21)

It can be proved that card(P(N*)) = c

22)

Tra i numeri cardinali maggiori di c, ricordiamo solo la "potenza del funzionale" ,

cioè il numero cardinale dell'insieme i cui elementi sono tutte le funzioni di R in R.

23)

L'insieme delle funzioni di dominio B a valori in A viene indicato col simbolo A^B.

Dati poi due numeri cardinali a, b, indichiamo con a^b il numero cardinale dell'insieme A^B, essendo A, B due insiemi qualsiasi tali che card(A)=a, card(B)=b (si può dimostrare che la def. è corretta).

Dato ora un qualunque insieme S, è facile dimostrare che il suo insieme delle parti P(S) è equipotente con l'insieme F delle funzioni .

Poichè la cardinalità dell'insieme {0, 1} è 2, le definizioni precedenti consentono di scrivere la formula

card (P(S)) = 2^{card
(S)}

Il Teorema 20) può allora essere riassunto nella formula

2^a>a (per qualsiasi numero cardinale a )

mentre il teorema 21) diventa

2^aleph⁰=c

24)

E' ancora oggetto di studi la seguente congettura ("Ipotesi del continuo"):

"Non esiste alcun numero cardinale compreso fra la potenza del numerabile e la potenza del continuo."

Si è riusciti a dimostrare (Cohen, 1963) che questa ipotesi non è nè dimostrabile, nè refutabile nella teoria di Zermelo-Fraenkel e nemmeno in quella di von Neumann (si tratta di teorie degli insiemi formalizzate …).

25)

Another really surprising statement could also be demonstrated:

THEOREM. -

The set of the points of a segment and the set of the points of a square, or of a cube, have the same cardinal number.

A segment, the straight line, the two-dimensional space, the three-dimensional space, have the same cardinal number.

… but I suppose it can be enough for now.

The proof of this proposition is omitted,

and our travel through the degrees of the infinite is over. Ain’t it been exciting?

Giancarlo Zilio

Via Vittorio Veneto, 12

13045 Gattinara (VC)

Tel. 0163-833518

e-mail gianzilio@libero.it

“Chi ha paura della matematica?”

http://www.chihapauradellamatematica.org/

yeeeaaahhh!!!