Summary:
+ natural numbers
+ Still Aleph-zero
+ The historical context
+ The Continuous
- The hypothesis of Continuous
c = aleph-1?
Issues ... theological!
Given the results in the previous chapter, it might seem hard to find infinite larger c or cardinal numbers (transfinite) are larger than c ... do exist, and how! I'm not entirely easy to find, but they were identified, however I can assure you that Cantor has shown that there are infinite numbers "transfinite", the first of which is Aleph-0, then there is Aleph-1 Aleph-2 and so on, creating a set ... infinite cardinal numbers transfinite!
c = aleph-1?
Now is a very interesting question. This c what is? We know it must be greater than Aleph-0: Aleph-1 coincide with mica, which is the number transfinite immediately after aleph-0?
We can ask the question another way. You will never find an infinite set has a power that is strictly between the sets of natural numbers and real numbers? That is, a structure which can not be counted by the real numbers, but can not in turn "counts" the continuous? If the answer is no, then we will automatically c = Aleph-1, or it may be c = Aleph-2 Aleph-3 or ...
Cantor has spent the last years of his life trying to prove that what mathematicians call "continuum hypothesis", ie that there is no insieme infinito compreso fra quelli dei numeri naturali e dei numeri reali, ma senza riuscirci.
Nel 1900 a Parigi, durante il Congresso Matematico Internazionale, David Hilbert (1862-1943) sottopose ai matematici di tutto il mondo un elenco di problemi da risolvere (il più noto forse è la dimostrazione dell’ultimo teorema di Fermat); il primo problema della lista era proprio dimostrare l’ipotesi del continuo.
Il problema si è rivelato davvero difficile da aggredire, tant’è che solo nel 1940 Kurt Gödel ha dimostrato che non è dimostrabile la falsità dell’ipotesi del continuo. Invece nel 1963 Paul Cohen ha dimostrato che è impossibile dimostrare che l’ipotesi del continuo sia vera...
... ecco quindi saltar fuori un’antinomia, un enunciato che porta a conclusioni contraddittorie! Solo che, a differenza del paradosso di Russell di cui ho parlato nei capitoli precedenti, che rivelava un’antinomia in una cosa che sembra (almeno a noi comuni mortali) più una pignoleria che un problema reale, adesso l’antinomia salta fuori nello studio dei numeri, cioè al livello più basilare di tutto ciò che è matematica!
A me questo risultato ha sempre riempito di meraviglia: quando l’ho letto la prima volta ho veramente fatto un salto sulla sedia. Mi sembra magnifico che l’intelletto umano sia riuscito a isolare un’antinomia only in studying the nature of the numbers!
The set of ideas that the characters I mentioned in these pages is able to put together is one of the highest peaks reached by humanity. It does not matter if the end result is an apparent "stalemate" with these contradictions that crop up everywhere. I totally agree with Carl Jacobi, who wrote:
"The only purpose of science is the honor of the human spirit, this title is a matter of numbers as a system in the world" The power oflogical processes initiated by Cantor with his approach to infinite sets has proved very formidable, though many mathematicians of his contemporaries did not immediately appreciated. Even in the midst of bitter disputes on this issue, David Hilbert said:
"No one will dispel the Paradise that Cantor has given us!"
Issues ... theological! About
Finally, we add a few anecdotes regarding Galileo and Cantor, struggling with this concept not only numerical but also philosophical and especially theological.
The concept of infinity is beginning to be exploited in theology by Nicholas of Cusa (1401-1464) that compares a number of occasions in the infinity of God with the finite nature of men, and the intellect (Finished) with the Truth (infinite). In the long run then the infinite has condensed a lot of divine attributes ... becoming a term to use with the springs.
As we have seen Galileo Galilei came across some logical reasoning concerning the infinite, but his lack of courage in reaching their ultimate consequences, perhaps is due to the fear of the Inquisition to other topics ... which had problems with its beautiful! Do not forget that a few years earlier, in 1600, Giordano Bruno was condemned to the stake for saying "the universe is therefore an infinite, while the second property ...", the Inquisition (Cardinal Robert Bellarmine is what is condemned Giordano Bruno, Galileo) and runs the universe is finite, while the earth to stand still. Even
Cantor poses some problems when deciding to disclose his views on the infinite, especially knowing the existence of an infinite number of different sizes. As a good Christian churches use the term if not infinite would disturb the ecclesiastical hierarchy was the end of the nineteenth century, there was no danger of going to the stake, but still wanted to know what he thought of this fact the Catholic Church. He went to the Vatican, brought his work to the Holy Office, which was ruled by a German cardinal, who said: "Eminence I have here works in mathematics I say there are countless more, many in fact infinite. " The cardinal said: "Well I do not know the math ... I let her work, I consider them to my secretary. "
The secretaries were the Dominicans, who took two years, because obviously they had to start studying set theory from scratch. After two years, the cardinal said: "In our opinion there is no problem, there is no danger to the faith." So Cantor was summoned to the Vatican and Cardinal of the Holy Office said: "Look you can talk about these endless, provided you call them infinite, because this actually darebbe una brutta idea teologica, cioè farebbe una connessione con la divinità". Allora Cantor scelse il nome "transfiniti". (Per ironia della sorte, i matematici preferiscono chiamarli con il nome di... numeri Cardinali!)
Il cardinale del Santo Uffizio si era anche fatto l’idea che dopo tutti questi transfiniti, là, alla fine, ci fosse il vero infinito assoluto. Chiese a Cantor cosa ne pensasse: "Per noi matematici quello non c'è. Non esiste un infinito assoluto per i matematici, perché ciò sarebbe contraddittorio". Al che il cardinale disse: "Va bene: quello lì è nostro!". ▲
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