Translated by: John Murphy








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L'ARITHMETIQUE constitue un SYSTEME FORMEL, un langage composé d'un certain nombre d'éléments. Les nombres entiers, puis des signes représentant les différentes opérations, le signe d'égalité =, l'implication =>, la négation, des choses comme (il existe), (pour tout)... Ajoutez à cela les AXIOMES DE PEANO, et c'est tout.
ARITHMETIC constitutes a FORMAL SYSTEM, a language composed of a certain number of elements. Whole numbers, then signs representing the various operations, the equals sign = , implication =>, negation, things such as (it exists), (for everything)… Add to that the PEANO AXIOM, and that's all

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L’ARITMETICA costituisce un SISTEMA FORMALE, un linguaggio composto da un certo numero di elementi. I numeri interi, poi i segni che rappresentano le differenti operazioni, il segno di uguaglianza =, l’implicazione => , la negazione, simboli come (esiste), (per tutto)… Aggiungete a questo gli ASSIOMI DI PEANO, ed è tutto

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Oui, je sais, il existe une présentation totalement axiomatisée de l'arithmétique
Yes I know, there exists a totally axiomatic presentation of arithmetic

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Sì, lo so, esiste una presentazione totalmente assiomatizzata dell’aritmetica

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Vous savez également qu'il existe une proposition indécidable qui se formule selon la question : est-ce que 0 est égal à 1 ? (*)
You also know that there exists an undecidable proposition which is formulated according to the question: is 0 equal to 1? (*)

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Sapete inoltre che esiste una proposizione irrisolvibile che si formula secondo la domanda: 0 è uguale ad 1? (*)

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Nous connaissons cette particularité, professeur Gödel, mais est-ce qu'il n'est pas possible d'inclure la proposition "0=1", considérée comme fausse, en tant que nouvel axiome de la THEORIE ARITHMETIQUE ?
We are aware of this particularity Professor Godel, but would it not be possible to include the proposition "0=1", considered as false, as a new axiom of ARITHMETICAL THEORY?

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Conosciamo questa particolarità, Professor Gödel, ma non è possibile includere la proposizione 0=1, considerata come falsa, come nuovo assioma della TEORIA ARITMETICA?

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(*) Aussi absurde que cela puisse paraître, dans ce contexte axiomatique il est impossible de démontrer ou de réfuter la proposition "0=1" La Direction
(*) As absurd as it may appear, in this axiomatic context it is impossible to demonstrate or refute the proposition "0=1" The Management

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(*) Per quanto assurdo possa sembrare, in questo contesto assiomatico è impossibile dimostrare o rifiutare la proposizione ˝0=1˝ La Direzione

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45

Hélas, mon jeune ami, je me suis aperçu que si on incluait ce nouvel axiome "zéro n'est pas égal à un", ceci engendrait aussitôt une nouvelle proposition indécidable, et ainsi de suite...
Alas, my young friend, I have noticed that if we include a new axiom "zero is not equal to one", this will immediately produce a new undecidable proposition, and so on…

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Ahimè, mio giovane amico, mi sono accorto che se si includesse questo nuovo assioma ˝zero non è uguale ad uno˝, ciò produrrebbe subito una nuova proposizione irrisolvibile, e così via…

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C'est proprement effrayant !
That's truly frightening!

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E’ davvero spaventoso!

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Il est donc impossible que l'arithmétique devienne jamais une THEORIE COMPLETE, CONSISTANTE, la proposition "0=1" devant être considérée comme ESSENTIELLEMENT INDECIDABLE
It is therefore impossible that arithmetic will ever become a COMPLETE CONSTANT THEORY, the proposition "0=1" must be considered as ESSENTIALLY UNDECIDABLE

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E’ quindi impossibile che l’aritmetica diventi una TEORIA COMPLETA, CONSISTENTE, in quanto la proposizione ˝0 = 1˝dev’essere considerata ESSENZIALMENTE IRRISOLVIBILE

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Bon, c'est bien triste. Mais il n'y a pas que l'arithmétique
That's a pity. But there isn't only arithmetic.

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Bene, è assai triste. Ma non c’è solo l’aritmetica.

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Est-ce que la solution ne consisterait pas à supprimer carrément le zéro, cause de tous ces maux ?
Couldn't a solution be the total suppression of zero, the cause of all our problems?

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La soluzione non consisterebbe nel sopprimere con decisione lo zero, causa di tutti questi problemi?

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Bernie, ces types sont fous
Bernie, these people are mad

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Bernie, questi tipi sono pazzi

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PHYSICIENS
PHYSICISTS

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FISICI

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Hélas...
Alas…

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Ahimé…

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Les propositions d'un langage sont associables aux nombres entiers de manière biunivoque et ce monde des nombres leur communique aussitôt cette maladie d'indécidabilité essentielle.
The propositions of a language are associable with whole numbers in a biunivocal manner and this world of numbers immediately communicates this essential undecidability to them.

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Le proposizioni di una lingua si possono associare ai numeri interi in modo biunivoco e questo mondo di numeri comunica loro subito questa malattia di irrisolvibilità essenziale.

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Professeur Gödel, je ne peux y croire
Professor Godel, I cannot believe this

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Professor Godel, non posso crederci.

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Ce défaut essentiel, tel un péché originel, préexistera donc dans TOUS les langages, dans la langue naturelle, les mathématiques...
This essential defect, like original sin, pre-exists in ALL languages therefore, in the natural language, in mathematics…

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Questo difetto essenziale, come un peccato originale, preesisterà quindi in TUTTE le lingue, nella lingua naturale, nel linguaggio matematico…

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46

Prenons par exemple le langage naturel construit à partir de signes comme les lettres de l'alphabet et considérons la suite des NOMBRES PREMIERS, excepté 1 : 2, 3, 5, 7, 11, 13, 17, 19, 23, etc... Nous allons considérer les lettres elles-mêmes comme des propositions et leur assigner un numéro. A a, la première lettre, nous associerons 21, à b, la seconde lettre, 22 = 4, à c 23 = 8, etc...
Take for example the natural language constructed from signs like the letters of the alphabet and let us consider the series of PRIME NUMBERS, except 1: 2, 3, 5, 7, 11, 13, 17, 19, 23 etc… We will consider the letters themselves as propositions and assign a number to them. We will assign 21 to a, the first letter, 2² = 4 to b, the second letter, 23 = 8 to c, and so on

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Prendiamo ad esempio il linguaggio naturale formato a partire da segni come le lettere dell’alfabeto e consideriamo la successione dei NUMERI PRIMI, eccetto 1: 2, 3, 5, 7, 11, 13, 17, 19, 23, ecc… Considereremo le lettere stesse come proposizioni ed assegneremo loro un numero. Ad a, la prima lettera, assoceremo 21 , a b, la seconda lettera, 22= 4, a c, 23= 8, ecc…

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Passons maintenant à une suite de caractères comme b-a-c. Nous allons cette fois baser le codage sur la suite des nombres premiers commençant par le 3, c'est-à-dire 3, 5, 7, etc... Le nombre associé à la lettre b étant 4, à la lettre a, 2 et à la lettre c, 8, formons le nombre entier n = 345278 égal à 11 673 722 025, qui est déjà grand. Je dis que ce nombre entier code parfaitement la suite de lettres b-a-c
Let us now move on to a series of characters such as b-a-e. This time we will base our coding on the series of prime numbers, beginning with 3, that is to say 3, 5, 7 etc… The number associated with the letter b being 4, with the letter a, 2 and the letter c, 8, we form the whole number n=345278 which equals 11 673 722 025, already high. I say that this whole number codes perfectly the series of letters b-a-e

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Passiamo ora ad una serie di caratteri come b-a-c. Questa volta baseremo la cifratura sulla serie dei numeri primi che iniziano dal 3, cioè 3, 5, 7, ecc… Essendo 4 il numero associato alla lettera b, 2 quello associato alla lettera a, ed 8 alla lettera c, formiamo il numero intero n = 345278 uguale a 11 673 722 025, che è già grande. Dico che questo numero intero codifica perfettamente la serie delle lettere b-a-c.

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Compris ! Ce nombre est DECOMPOSABLE DE FACON UNIQUE selon un produit de puissances entières de nombres premiers. Ce qui fait que de n = 11 673 722 025 on retourne au produit 345278. On peut alors décomposer les exposants
Understood! This number is DECOMPOSABLE IN A UNIQUE WAY according to the product of whole powers of prime numbers. Which makes it that from n = 11 673 722 025 we return to the product 345278. We can then decompose the exponents

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Capito! Questo numero è DECOMPONIBILE IN MODO UNICO secondo un prodotto di potenze intere di numeri primi. Ciò fa sì che da n = 11 673 722 025 si ritorni al prodotto 345278. Possiamo allora scomporre gli esponenti

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47

Par quel miracle, grands dieux !
By what miracle for goodness' sake?

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Per quale miracolo, santi numi!

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Prends ce nombre, qui vaut 11 673 722 025. Si tu le divises par les nombres premiers, encore et encore, jusqu'à ce que cela ne soit plus possible, tu obtiens une décomposition selon un produit de puissances entières de nombres premiers. Ici 345278. Cette décomposition est UNIQUE.
Take this number, which equals 11 673 722 025. If you divide it by prime numbers again and again, until it is no longer possible, you obtain a decomposition according to a product of the whole powers of prime numbers. Here 345278. This decomposition is UNIQUE

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Prendi questo numero, che vale 11 673 722 025. Se lo dividi per i numeri primi ancora e ancora, fino a che questo non sia più possibile, ottieni una scomposizione secondo un prodotto di potenze intere di numeri primi. Qui 345278. Questa scomposizione è UNICA.

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En prenant la puissance du premier nombre premier de la série, 34, et en considérant cet exposant comme une puissance de 2 on retrouve le rang du premier caractère du mot, soit 2, qui correspond à la lettre b
By taking the power of the first prime number of the series, 34, and in considering this exponent as a power of 2, we find the rank of the first character of the word, that is 2, which corresponds to the letter B

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Prendendo la potenza del primo numero della serie, 34, e considerando questo esponente come una potenza di 2 si ritrova il rango del primo carattere della parola, cioè 2, che corrisponde alla lettera b

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Grâce à cette méthode de codage le nombre 11 673 722 025 représente le mot bac et uniquement celui-là
By this method of coding, the number 11 673 722 025 represents the word bae and only that

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Grazie a questo metodo di codificazione il numero 11 673 722 025 rappresenta la parola bac e solo quella

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TROIS NE DIVISE…
THREE IS NOT DIVISIBLE…

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TRE NON DIVIDE…

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En codant le "blanc" entre les mots, ou la ponctuation, on peut associer un nombre à une suite de mots, à une phrase complète
By coding the "space" between the words, or punctuation, we can associate a number with a series of words, with a whole phrase

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Codificando il ˝bianco˝tra le parole, o la punteggiatura, possiamo associare un numero ad una serie di parole, ad una frase completa

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48

Les phrases d'une langue quelconque peuvent être représentées par un ensemble de nombres entiers assez longs
The phrases of any language can be represented by a set of fairly long whole numbers

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Le frasi di una qualsiasi lingua possono essere rappresentate da un insieme di numeri interi abbastanza lunghi

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