Translated by: John Murphy








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PARTITION : d'un côté LES OUVRAGES QUI SE CITENT EUX-MEMES et de l'autre LES OUVRAGES QUI NE SE CITENT PAS EUX-MEMES
Of course. But SET THEORY has a response to EVERYTHING. We create a new PARTITION: on one side WORKS WHICH CITE THEMSELVES and on the other WORKS WHICH DO NOT CITE THEMSELVES

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Certo. Ma avendo la TEORIA DEGLI INSIEMI una risposta a TUTTO, CREEREMO UNA NUOVA partizione: da un lato LE OPERE CHE CITANO SE’ STESSE e dall’altro LE OPERE CHE NON CITANO SE’ STESSE.

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Ceci permet de classer TOUS les ouvrages existants. Impressionnant
This allows the classification of ALL existing works. Impressive.

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Ciò consente la classificazione di TUTTE le opere esistenti. Impressionante.
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OUVRAGES QUI SE CITENT EUX-MEMES
WORKS WHICH CITE THEMSELVES

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OPERE CHE CITANO SE’ STESSE.

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OUVRAGES QUI NE SE CITENT PAS EUX-MEMES
WORKS WHICH DO NOT CITE THEMSELVES

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OPERE CHE NON CITANO SE’ STESSE.

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Et je suppose qu'on peut cataloguer ces ouvrages ?
And I suppose that we can catalogue these works?

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E suppongo che si possano catalogare queste opere?

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On peut TOUT cataloguer
We can catalogue EVERYTHING

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Si può catalogare TUTTO.

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Excusez une dernière question naïve, où mettriez-vous, parmi ces deux ensembles, L'OUVRAGE QUI CITERAIT TOUS CEUX QUI NE SE CITENT PAS EUX-MEMES ET EXCLUSIVEMENT CEUX-LA !
Excuse me for a final, naive question, where would you put WORKS WHICH CITE ALL THOSE THAT DO NOT CITE THEMSELVES AND ONLY THOSE among the two groups?

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Scusate un’ultima domanda ingenua, dove mettereste, tra questi due insiemi, L’OPERA CHE CITEREBBE TUTTI QUELLI CHE NON SI CITANO ED ESCLUSIVAMENTE QUELLI!

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14

Voyons, cet ouvrage doit-il se citer ? S'il se cite, c'est qu'il ne décrit plus EXCLUSIVEMENT les ouvrages qui ne se citent pas eux-mêmes, puisqu'il cite un ouvrage qui se cite, à savoir... LUI-MEME...
Let's see, should this work cite itself? If it does cite itself then it no longer EXCLUSIVELY describes works that do not cite themselves, because it cites a work that does cite itself, that is… ITSELF

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Vediamo, quest’opera deve citarsi? Se si cita, accade che non descrive più ESCLUSIVAMENTE le opere che non si citano, dato che cita un’opera che si cita… che è… SE’ STESSA…

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Alors il ne doit pas se citer. Mais dans ces conditions, étant censé citer TOUS les ouvrages qui ne se citent pas eux-mêmes, il en omettrait un : LUI-MEME...
So it shouldn't cite itself. However under those conditions, as it is supposed to cite EVERY work that does not cite itself, one will be omitted…THIS WORK

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Allora essa non deve citarsi. Ma in queste condizioni, supponendo di citare TUTTE le opere che non si citano, se ne ometterebbe una: QUEST’OPERA…

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AHHRGL !
AAARGH!

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AHHRGH! 

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Quoi... quoi... qu'est-ce que j'ai dit ?
What… what… what did I say?

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Cosa… cosa… cosa ho detto?

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CHPAF !
SHPLAT!

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CHPAF!

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Il a salement disjoncté
He's cracked completely

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E’andato in corto circuito.

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Je n'y suis pour rien...
It's not my fault…

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Io non c’entro…

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15

EXIT LA THEORIE DES ENSEMBLES en tant que FONDEMENT DU LANGAGE
SO EXIT SET THEORY as a FOUNDATION OF LANGUAGE

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NASCE LA TEORIA DEGLI INSIEMI come FONDAMENTO DEL LINGUAGGIO

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De la fumée sort par ses oreilles
Smoke is coming out of his ears

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Dalle sue orecchie esce del fumo

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EXTINCTEUR
EXTINGUISHER

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ESTINTORE

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Oui, je vois. Tiens le moi une seconde
Yes, I see. Hold this a second for me

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Sì, vedo. Tienimelo un secondo.

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Ca sort de l'autre côté !
It's coming out of the other side!

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Esce dall’altra parte!

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C'est normal ! Ca prouve que ça marche
It's normal! It proves that it works

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E’ normale! Questo prova che funziona.

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Ca va mieux ?
Feeling better?

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Va meglio?

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Oui... merci...
Yes… thanks…

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Sì… grazie…

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Je suggère que nous allions voir les MATHEMATICIENS, c'est bien le diable s'ils n'ont pas une recette pour nous tirer de là...
I suggest that we go and see MATHEMATICIANS, I'd be amazed if they haven't got a recipe to get us out of this.

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Suggerisco di andare a vedere i MATEMATICI, sarebbe strano se non avessero una ricetta per aiutarci…

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Ma question de tout à l'heure semble avoir déclenché chez ce pauvre homme un état de stress intellectuel intense. J'avoue que je n'ai pas très bien compris pourquoi...
My question earlier seems to have caused a state of intense intellectual stress for this poor gentleman. I admit that I'm not sure why…

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La mia domanda di prima sembra aver causato a questo povero uomo uno stato d’intenso stress intellettuale. Confesso di non aver capito bene perché…

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C'est simple : il parlait du principe que l'on peut toujours classer les choses, les rattacher à des ensembles DEFINIS PAR DES PROPRIETES
It's simple: he was talking about the principle whereby things can always be classified, attached to groups DEFINED BY PROPERTIES

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E’ semplice: parlava del principio che si possono sempre classificare le cose, ricollegarle a degli insiemi DEFINITI DA DELLE PROPRIETA’

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16

Voici une autre façon de formuler ce PARADOXE. Imagine un village où l'on classe les hommes en deux ensembles, selon le CRITERE D'APPARTENANCE suivant. L'un contient les hommes qui se rasent eux-mêmes et l'autre ceux qui ne se rasent pas eux-mêmes. Imagine un barbier dont la fonction soit de raser tous les hommes qui ne se rasent pas eux-mêmes, et EXCLUSIVEMENT ceux-là
Here's another way to formulate this PARADOX. Imagine a village where men are classed into two groups according to the following CRITERION OF BELONGING. One contains the men who shave themselves and the other those who do not shave themselves. Imagine a barber whose job it is to shave all those men who do not shave themselves, and EXCLUSIVELY those

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Ecco un altro modo di formulare questo PARADOSSO. Immagina un villaggio in cui gli uomini sono classificati in due insiemi, secondo il seguente CRITERIO D’APPARTENENZA. Uno contiene gli uomini che si rasano, l’atro quelli che non si rasano. Immagina un barbiere il cui compito sia quello di rasare tutti gli uomini che non si rasano, ed ESCLUSIVAMENTE quelli.

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Question : à quel ensemble appartient ce barbier ?
Question: which set does the barber belong to?

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Domanda. A quale insieme appartiene questo barbiere?

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?
?

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?

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Voyons, s'il se rase, c'est qu'il rase une personne qui se rase elle-même, ce qui est contraire à sa fonction
Let's see, if he shaves himself then he is shaving a person who shaves themself, which is contrary to his function.

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Vediamo, se lui si rasa, significa che rasa una persona che si rasa, che è contrario al suo compito.

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Mais s'il ne se rase pas, c'est qu'il omet de raser une personne qui ne se rase pas elle-même. Dilemme...
But if he doesn't shave himself, then he has omitted to shave a person who does not shave themself. Dilemma…

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Ma se lui non si rasa, significa che omette di rasare una persona che non si rasa. Dilemma…

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En vérité tous les langages sont bourrés de contradictions de ce genre. La PROPOSITION "je mens" n'est ni vraie... ni fausse !
In truth, all languages are full of contradictions like this. The PROPOSITION "I lie" is neither true… nor false!

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In verità tutte le lingue sono piene di contraddizioni di questo tipo. La PROPOSIZIONE “io mento” non è né vera…né falsa!

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C'est une proposition INDECIDABLE
It's a propostition that is UNDECIDABLE

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E’ una proposizione IRRISOLVIBILE

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17

ENSEMBLES NON CANTORIENS (*)
NON-CANTORIAN GROUPS (*)

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GRUPPI NON-CANTORIANI (*)

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Le malaise qui sévit actuellement trouverait peut-être sa racine dans ces défauts des fondements même des langages
Maybe the current malaise stems from defects in the very bases of the language

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Il malessere attuale forse trova il proprio fondamento in queste imprecisioni delle basi del linguaggio

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Des mots, toujours des mots. Parlez-moi de la pureté des nombres de des FORMALISMES MATHEMATIQUES
Words, always words. Talk to me about the purity of the numbers in MATHEMATIC FORMALISMS

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Parole, sempre parole. Parlatemi della purezza dei numeri e dei FORMALISMI della MATEMATICA

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Si nous allions faire un tour à l'Institut de Numérologie
What if we took a trip to the Numerology Institute?

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Se facessimo un giro all’Istituto di Numerologia?

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En route !
Let's go!

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Andiamo!

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On arrive bientôt...
We'll soon be there

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Siamo quasi arrivati…

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Ecoutez !
Listen!

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Ascoltate!

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...éminent spécialiste de la THEORIE DES ENSEMBLES vient de se suicider par le feu sur la tombe d'Evariste Galois (**)
…eminent specialist in SET THEORY has just killed himself by fire on the tomb of Evariste Galois (**)

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…eminente specialista della TEORIA DEGLI INSIEMI si è appena suicidato col fuoco sulla tomba di Evariste Galois (**)

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Qu'est-ce qu'ils sont en train de faire ?
What are they doing now?

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Cosa stanno facendo?

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Cela ressemble à une passoire avec une lampe électrique dedans ?!?
It looks like a sieve with an electric lamp inside?!?

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Assomiglia ad uno scolapasta con una lampada elettrica dentro?!?

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Descend un peu la lampe, doucement...
Lower the lamp a little, gently…

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Abbassa un po’ la lampada, dolcemente…

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(*) du nom du mathématicien allemand Georg Cantor (1845-1918), créateur de la théorie des ensembles
(*) from the name of the German mathematician Georg Cantor (1845-1918), creator of set theory

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(*) dal nome del matematico tedesco Georg Cantor (1845-1918), fondatore della teoria degli insiemi

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(**) mathématicien français (1811-1832). Tué dans un duel
(**) French mathematician (1811-1832). Killed in a duel

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(**) matematico francese (1811-1832). Ucciso in un duello.

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18

Vas-y, allume
Go on, turn it on

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Dai, accendi.

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Merveille des merveilles ! Ca marche !
Marvel of marvels! It works!

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Meraviglia delle meraviglie! Funziona!

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Mais qu'est-ce que vous fabriquez, sapristi ?
But what are you up to for heaven's sake?

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Ma cosa state combinando?

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Vous ne VOYEZ pas ? Grâce à ce système à chaque trou du plan je peux faire correspondre un trou ménagé dans la passoire
Can't you SEE? Thanks to this system I can make each hole in the plane correspond to a hole made in the sieve.

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Non VEDETE? Grazie a questo sistema ad ogni foro del piano posso far corrispondere un foro disposto nello scolapasta.

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Mais, un plan possède une infinité de points ?
But a plane possesses an infinity of points?

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Ma un piano possiede un’infinità di punti?

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P P'
P P'

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P P’

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Il suffit de prendre une passoire avec une infinité de trous
Well you just need a sieve with an infinity of holes

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Basta prendere uno scolapasta con un’infinità di fori

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De la même manière on peut relier deux à deux l'infinité des points d'une droite à l'infinité des points présents sur un arc de cercle
In the same way we can link two by two the infinity of points in a straight line to the infinity of points present in the arc of a circle

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Allo stesso modo è possibile unire due a due l’infinità dei punti di una linea retta all’infinità dei punti presenti nell’arco di un cerchio

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19

Comprenez-vous la transcendance de cette découverte ? Il y a une infinité de points sur une droite et il y a également une infinité de points sur un demi-cercle. Mais l'expérience précédente montre qu'il y en a AUTANT. De même il y en a autant sur un plan infini que sur une demi-sphère, ou un disque
Do you understand the transcendance of this discovery? There is an infinity of points on a straight line and, equally, an infinity of points on a semi-circle. But the preceding experiment shows that there are AS MANY. The same as there are as many on an infinite plane as on a hemisphere, or a disc

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Capite la trascendenza di questa scoperta? Ci sono un’infinità di punti su una linea retta e ci sono ugualmente un’infinità di punti su un semicerchio. Ma il precedente esperimento mostra che ce ne sono ALTRETTANTI. Analogamente ce ne sono altrettanti sia su un piano infinito che su una mezza sfera, o un disco

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Avec sa passoire, il a capturé l'infini
With his sieve he's caught infinity

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Col suo scolapasta, ha catturato l’infinito

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Mais enfin, l'infini, c'est l'infini, non ?
But hey, infinity is infinite isn't it?

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Ma alla fine, l’infinito, è l’infinito, no?

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On peut comparer deux infinis entre eux, vérifier s'ils sont COMMENSURABLES. D'après ce qui précède on voit que la droite et le demi-cercle ont le même nombre de points. Il en est de même pour le plan et la demi-sphère.
We can compare two infinities with each other, verify if they are COMMENSURABLE. After what we have just seen, we can see that the straight line and the half circle have the same number of points. It's the same for the plane and the hemisphere.

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E’ possible comparare due infiniti tra loro, verificare se sono commensurabili. Dopo ciò che abbiamo visto si nota che la linea retta ed il semicerchio hanno lo stesso numero di punti. E’ lo stesso per il piano e la mezza sfera.

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Déformons maintenant le demi-cercle selon un segment de droite
Now let's deform the semi-circle according to a segment of a straight line

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Deformiamo ora il semicerchio secondo un segmento della linea retta.

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Attendez... Le segment est lui-même une partie de la droite. Ainsi, quand on raisonne sur l'infini, LA
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