Russell's paradox

Russell's paradox or barber's paradox, credited to Bertrand Russell, shows that the original set theory formulated by Cantor and Frege is contradictory.

The paradox in terms of sets

Suppose the cases of sets that are members of themselves. An example described is one involving a set consisting of "abstract ideas". This set is a member of itself because the set itself is an abstract idea. Another example would be a bag with bags inside. On the other hand a set consisting of "books" is not a member of itself because the set itself is not a book. Russell asked (in a letter written to Frege in 1902), if the set of sets that are not part of themselves (that is, the set that encompasses all those sets that are not included in themselves, such as &# 34;books" in the example above) is part of itself. The paradox is that if it is not part of itself, it belongs to the type of sets that are not part of themselves and therefore is part of itself. That is, it will be part of itself only if it is not part of itself.

Formal statement of the paradox

Call. M{displaystyle M_{}^{}{}} the "set of all sets that do not contain themselves as members." I mean

(1)M={x:x x!{displaystyle M={x:xnotin x}}

According to Cantor's set theory, equation (1) can be represented by

(2)Русский Русский xx한 한 M x x{displaystyle forall xqquad xin Miff xnotin x}

that is, "Every set is an element of M{displaystyle M} yes and only if it is not an element of itself."

Now, in view of that M{displaystyle M} is a set, can be replaced x{displaystyle x} for M{displaystyle M} in the equation (2), where it is obtained

(3)M한 한 M M M{displaystyle Min Miff Mnotin Mnotin M}

I mean, M{displaystyle M} is an element of M{displaystyle M} Yes and only if M{displaystyle M} is not an element of M{displaystyle M}Which is absurd.

The paradox in terms of the barber

Russell's paradox has been expressed in various more everyday terms, the best known being the barber's paradox which can be stated as follows:

In a distant village of an ancient emirate there was a barber named As-Samet diestro in shaving heads and beards, master in scamonding feet and putting leeches. One day the emir realized the lack of barbers in the emirate, and ordered that the barbers only shave those who could not shave themselves. Ah! and imposed the rule that everyone was shaved, (it is not known whether by hygiene, by aesthetics, or by demonstrating that he could impose his holy will and thus show his power). One day the emir called As-Samet to shave him and he told him his anguish: "In my village I am the only barber. I cannot shave the barber of my people, that I am!, for if I do, then I can shave myself, therefore I should not shave! for I would disobey your order. But, if on the contrary I do not shave, then some barber should shave me, but as I am the only barber there!, I cannot do it and also so disobey you my lord, O emir of believers, may Allaah have you in his glory! The emir thought that his thoughts were so deep, he anticipated it with the hand of the most beautiful of his concubines. Thus the barber As-Samet lived forever happy and barbon.

In first-order logic, the barber's paradox can be expressed as:

(4)Русский Русский xafeita(x,barberor) ¬ ¬ afeita(x,x){displaystyle forall xqquad mathrm {afeita} (x,barbero)iff neg mathrm {afeita} (x,x)}

Where afeita(x,and){displaystyle mathrm {afeita} (x,y)} means "x{displaystyle x} is shaved by and{displaystyle and}". The above would be read as "Every person is shaved by the barber if and only if he does not shave himself." It is important to note the similarity between equations (2) and (4). Al substitution x{displaystyle x} for barberor{displaystyle barbero} obtained

(5)afeita(barberor,barberor) ¬ ¬ afeita(barberor,barberor){displaystyle mathrm {afeita} (barber, barber)iff neg mathrm {afeita} (barber, barber)}

That is, the barber shaves himself if and only if he doesn't shave himself, which is a contradiction.

But Russell doubts this formulation, he himself comments “On one occasion a formulation was suggested to me that was not valid; namely, the question of whether or not the barber shaves himself. You can define the barber as "someone who shaves all those, and only those, who do not shave themselves." The question now is: does the barber shave himself? Thus formulated, the contradiction is not very difficult to resolve.

Explanation of the paradox

Sets are collections of things, for example cars, books, people, etc. and in this sense we will call them normal sets.

The main characteristic of a normal set is that it does not contain itself. But there are also setups, such as 2M{displaystyle 2^{M}}, which is the set of subsets of M.

A set of sets is normal unless we can make it contain itself. The latter is not difficult if we have the set of all things that are NOT books and since a set is not a book, the set of all things that are NOT books will be part of the set of all things that are NOT books. These self-contained sets are called singular sets.

It is clear that a given set is either normal or singular, there is no middle term, or it contains itself or is not contained. Now let's take the set C{displaystyle C} as the whole of all normal sets. What kind of set is C{displaystyle C}Norm or Singular?

If it's normal, it'll be within the normal setup, which is C{displaystyle C}, then it can no longer be normal, since it contains itself. If it's singular, it can't be within the normal setup, then it can't be in C{displaystyle C}But if he can't be in C{displaystyle C} then it is not singular, since it does not contain itself.

History

Russell discovered the paradox in May or June 1901. According to his own account in his 1919 Introduction to Mathematical Philosophy, "I tried to discover some flaw in Cantor's proof that there is no cardinal (number) greater than all others". In a 1902 letter, he announced the discovery to Gottlob Frege of the paradox in Frege's 1879 Begriffsschrift and framed the problem in terms of both logic and set theory, and in particular in terms of Frege's definition of a function:

There's only one point I've met with difficulty. You say (p. 17 [p. 23 above]) that a function can also act as the undetermined element. This I believed earlier, but now this point of view seems doubtful because of the following contradiction. Let go w be the preacher: to be a preacher who cannot be preached of himself. Power w to be preached of himself? From each answer, the opposite follows. We must therefore conclude that w He's not a preacher. Likewise, there is no class (as a whole) of those classes that, taken as a whole, do not belong to themselves. From this I conclude that under certain circumstances a definable collection [Menge] does not form a whole.

Russell would go on to cover it extensively in his 1903 The Principles of Mathematics, where he repeated his first encounter with the paradox:

Before we say goodbye to fundamental questions, it is necessary to examine in more detail the singular contradiction, already mentioned, regarding the unpredictable preachers themselves.... I can mention that I was led to him in the effort to reconcile the Cantor test. "

Russell wrote Frege about the paradox just as Frege was preparing the second volume of his Grundgesetze der Arithmetik. Frege responded to Russell very quickly; his letter of June 22, 1902 appeared, with van Heijenoort's commentary in Heijenoort 1967: 126–127. Frege then wrote an appendix admitting the paradox, and proposed a solution which Russell would support in his Principles of Mathematics, but was later found unsatisfactory by some. For his part, Russell had his work in the printing press and added an appendix on the doctrine of types.

Ernst Zermelo in his (1908) A New Proof of the Possibility of Good Ordering (published at the same time as he published "the first axiomatic theory of sets") claimed the previous discovery of the antinomy in Cantor's naive set theory. He states: "And yet, even the elementary form that Russell⁹ gave to set-theoretic antinomies might have persuaded them [J. König, Jourdain, F. Bernstein] that the solution to these difficulties should not be sought in renouncing good order but only in an adequate restriction of the notion of set". Footnote 9 is where he makes his statement:

⁹1903, pp. 366-368. However, I myself had discovered this antinomy, regardless of Russell, and had communicated it before 1903 to Professor Hilbert, among others.

Frege sent a copy of his Grundgesetze der Arithmetik to Hilbert; as noted above, the last volume of Frege mentions the paradox that Russell had communicated to Frege. After receiving the last volume from Frege, on November 7, 1903, Hilbert wrote a letter to Frege in which he said, referring to Russell's paradox: "I think Dr. Zermelo discovered it three or three years ago." four years". A written account of Zermelo's actual plot was discovered in Edmund Husserl's Nachlass.

In 1923, Ludwig Wittgenstein proposed "undoing" Russell's paradox as follows:

The reason a function cannot be its own argument is that the sign of a function already contains the prototype of its argument, and It cannot contain itself. Let us suppose that the F(fx) function could be its own argument: in that case there would be a proposition F(F(fx))in which the outer function F and internal function F must have different meanings, since the interior has the form O(fx) and the outside has the shape Y(O(fx). Only the letter 'F' is common to both functions, but the letter alone means nothing. This is immediately clear if instead of F(Fu) We write (do): F(Ou). Ou = Fu. That eliminates Russell's paradox. (Tractatus Logico-Philosophicus3,333)

Russell and Alfred North Whitehead wrote their "Principia Mathematica" in three volumes in the hope of achieving what Frege had been unable to do. They tried to banish the paradoxes of naive set theory by employing a theory of types that they devised for this purpose. While they did manage to substantiate arithmetic in some way, it is not entirely evident that they did so by purely logical means. While "Principia Mathematica" it avoided the known paradoxes and allowed the derivation of a great deal of mathematics, the system of which gave rise to new problems.

In any case, Kurt Gödel in 1930–31 showed that while the logic of much of "Principia Mathematica", now known as first-order logic, is complete, Peano arithmetic is necessarily incomplete if consistent. This is widely, though not universally, regarded as having shown Frege's logician program impossible to complete.

In 2001, an International Centennial Conference celebrating the first hundred years of Russell's paradox was held in Munich and its proceedings were published.

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