Sommario:
+ I numeri Naturali
+ Ancora Aleph-zero
– Il contesto storico
Rifondare la matematica
Giuseppe Peano
Bertrand Russell
Gottlob Frege
Kurt Gödel
Irrationality of whole roots
+ + The Continuous
the continuum hypothesis
reshaping the mathematical
In the nineteenth century mathematics had reached a remarkable level of development, and was in constant evolution. Someone began to have doubts about the correctness of this, because it is true that conditions apply, with correct reasoning, it always comes to conclusions as valid, but we are sure that all the assumptions are valid, and that somewhere is not have made a few mistakes of reasoning?
So here he says there were two kinds of doubts. The first, already mentioned by Kant in "Critique of Pure Reason", was, as the process of induction can be accomplished without "off on a tangent? The second: the foundation upon which rests the whole castle are really solid?
The Mathematics was (and is) a field key, allowing to describe the behavior of physical systems. In this field you play casually with enough zeros, infinite and infinitesimal, become even infinite sums of infinitesimals. Fortunately, the analysis is often the way to deal with the physics, which historically has also happened that was not the physical to be "understood" by mathematics, but mathematics to be "validated" by physics.
course, this approach was not enough to mathematicians, fussy and perfectionist as I am here, therefore, necessary to make a comprehensive argument on the matter, trying to start from the smallest number of axioms, and then get to the more complex issues for small steps , incontrovertibly dimostrabili. (Gli assiomi, anche detti postulati, sono enunciati che, pur non essendo dimostrati, sono considerati veri; vengono usati per fornire i punti di partenza necessari alla delineazione di un quadro teorico, come può essere quello della teoria degli insiemi). ▲
Giuseppe Peano
Dovendo partire dalle cose più semplici, la cosa ovvia era di partire dai numeri naturali. I quali, pur essendo intuitivi, richiedevano una definizione precisa. Giuseppe Peano (1858-1932), il primo logico matematico italiano della storia, called them on the basis of these five axioms:
- There is a natural number zero
- Every natural number has a natural number successor
- different numbers have different successors
- Zero is not the successor of any natural number
- Each set of natural numbers containing zero and the successor of each element coincides precisely with the entire set of natural numbers
seems to me a wonderful definition, but did not appeal to everyone. Why, for example, the definition of "successor" was not considered sufficiently precise, you can count by twos or threes in three ... (i logici, coloro che si occupano di logica, riescono ad essere addirittura più puntigliosi e precisi dei matematici!)
I logici sanno che devono stare molto attenti: non è detto che le loro costruzioni logiche non entrino mai in contraddizione! Lo scoprirono già gli antichi greci, grazie a tale Epimenide di Creta (VI secolo a.C.), il quale, cretese appunto, ebbe a dire che "tutti i Cretesi sono bugiardi". Si capisce bene che si tratta di un paradosso: se egli, cretese, stesse dicendo la verità, allora non sarebbe vero che tutti i cretesi sono bugiardi; se invece fosse bugiardo, starebbe affermando una cosa vera! (*) Contraddizioni di questo tipo, o "antinomie" (le situazioni per cui, posta una particular issue, if they can make two statements apparently valid but which are contrary to each other) are always lurking.
(*) The paradox of the liar, as expressed by Epimenides, is incorrect: just think that at least one Cretan is telling the truth. Epimenides is a liar then actually saying that all Cretans are liars, because in reality there is one that is not. However, the modern logicians have been able to create paradoxes so other ... there is no loophole that takes! ▲
Bertrand Russell
So we think the logic a bit 'up, found that the natural numbers are a concept of "inappropriate", and looking for something more powerful for their meditations: invented the concept of set. Now I have tried several times to read some parts of the "Principia Mathematics" by Bertrand Russell (1872-1970), one of the greatest mathematical logicians ever existed, I say that I've just tried, for accuracy, the endless distinctions, I would say the pedantry are brought to levels ... that are not made for us mere mortals! Then, just to give you an example of what are the arguments of these gentlemen are talking about logic, and risk of saying something not quite right, try to explain how they can be seen the natural numbers.
Sets are collections of one or more distinguishable from one another (there is also the empty set), the number of items from each set is called the power of the whole. The empty set has zero energy and then, those with a single element has a power, and so on. To obtain a full set of natural numbers, just follow these steps:
- the empty set, which has zero elements and thus has power 0, we associate the number zero.
- we create the rule that the successor of a set A B is given by the union of elements of A with the same set A, so I add the element consists of a set A to elements already contained in A, obtaining a set B which has a capacity greater than the whole unit to: Association for the power this new set the corresponding natural number.
We see better how it works: the elements of the empty set (which has none) I add the empty set itself, by getting a set that contains only an empty set, and then have a power, the elements of ' a set (which contains an empty set) add a whole and get the two together (the empty set and will set a), the three will have three elements (The empty set, one and two), and so on, each with power equal to the number of items it contains. Having associated with each of these sets of numbers that express their power, with this we have defined a full set of natural numbers.
In essence, it takes the concepts of set (and the empty set) and successor to do all the work!
Among the nineteenth and twentieth century studies to re-establish the mathematics has been proceeding apace, with and without the use of set theory (not all the mathematicians saw her favorably, but as we have seen, and we'll see still, Georg Cantor used it in a spectacular way to attack the concept of infinity). Bases hours were really solid, and there was widespread belief that he would never have found a contradiction, would never have found a paradox (or antinomy, like that of the liar). ▲
Gottlob Frege
Even Gottlob Frege (1848-1925), considered one of the greatest logicians since Aristotle, had been helping reconstruction of mathematics. He had already published the first volume of his "Principles of Arithmetic" and was about to go to press with the second volume, she receives a letter from Bertrand Russell. Russell faces la seguente questione:
Può un insieme essere elemento di sé stesso, ovvero contenere se stesso?La risposta è sì. Ad esempio, l'insieme di tutti i libri di una biblioteca non è elemento di sé stesso. Invece, l'insieme di tutti gli insiemi con più di 20 elementi è elemento di sé stesso. Allora vediamo quest’altra questione:
Che tipo di insieme salta fuori se ne creo uno che contenga tutti gli insiemi che non contengono se stessi?Vediamo per tentativi, provando a considerare o meno questo insieme come elemento di se stesso:
— se dico che questo insieme non contiene se stesso, fa parte del gruppo degli insiemi che non contengono se stessi, e quindi dovrebbe be part of it (but the assumption was that it was part)
- if I say that this set contains itself, is not part of the group of sets that do not contain themselves, and therefore should not be part of it (but the ' hypothesis was that it was part)
fatal is the paradox: the creation of all the sets that do not contain themselves follows the appearance of a contradiction, and it is enough to dismantle the illusion of a logical system is complete and consistent . The existence of a contradiction as this is the crack that destroys the castle.
Frege took note of the destructive consequences for the system he had built and he resigned himself to write an addendum to its principles, in which he confessed the failure of his work. The contradictions highlighted by Russell's paradox is insoluble within the set theory, mathematicians and logicians have had a hard enough time learning how to handle this situation. ▲
Kurt Gödel
The illusion of a castle built entirely consistent mathematics was finally broken in 1931 when Kurt Gödel (1906-1978) proved his first incompleteness theorem. This says
In any mathematical theory T ... esiste una formula F tale che, se T è coerente, allora né F né la sua negazione sono dimostrabili in T .Questo teorema (semplificando) afferma che in un sistema assiomatico salterà sempre fuori un enunciato non dimostrabile a partire dagli assiomi di partenza, ovvero un caso indecidibile del quale non si può dire se sia vero oppure falso: e qui torniamo al paradosso del mentitore... che i greci avessero già capito tutto?
Il Secondo Teorema di Incompletezza recita invece:
Nessun sistema coerente può essere utilizzato per dimostrare la sua stessa coerenza.In pratica, se voglio costruire un sistema matematico starting from some axiom of departure, I will need some external axiom to the theory in order to verify the validity ... which I do not know if it makes much sense!
But be careful, at this point we do not say that mathematics is "all wrong" means the risk of throwing out the baby with the bathwater! It was discovered that the only logical consistency, when you reach certain limit reasoning, is not fully accessible. But when you pay the bill dell'Ortolano, rest assured that in the calculation of the rest do not present any contradiction! ▲
Next Chapter: The Continuous
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