Russell's paradox - definizione. Che cos'è Russell's paradox
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Cosa (chi) è Russell's paradox - definizione

PARADOX IN SET THEORY CONCERING THE SET OF ALL SETS NOT CONTAINING THEMSELVES
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Russell's Paradox         
<mathematics> A logical contradiction in set theory discovered by Bertrand Russell. If R is the set of all sets which don't contain themselves, does R contain itself? If it does then it doesn't and vice versa. The paradox stems from the acceptance of the following axiom: If P(x) is a property then x : P is a set. This is the Axiom of Comprehension (actually an axiom schema). By applying it in the case where P is the property "x is not an element of x", we generate the paradox, i.e. something clearly false. Thus any theory built on this axiom must be inconsistent. In lambda-calculus Russell's Paradox can be formulated by representing each set by its characteristic function - the property which is true for members and false for non-members. The set R becomes a function r which is the negation of its argument applied to itself: r = x . not (x x) If we now apply r to itself, r r = ( x . not (x x)) ( x . not (x x)) = not (( x . not (x x))( x . not (x x))) = not (r r) So if (r r) is true then it is false and vice versa. An alternative formulation is: "if the barber of Seville is a man who shaves all men in Seville who don't shave themselves, and only those men, who shaves the barber?" This can be taken simply as a proof that no such barber can exist whereas seemingly obvious axioms of set theory suggest the existence of the paradoxical set R. Zermelo Frankel set theory is one "solution" to this paradox. Another, type theory, restricts sets to contain only elements of a single type, (e.g. integers or sets of integers) and no type is allowed to refer to itself so no set can contain itself. A message from Russell induced Frege to put a note in his life's work, just before it went to press, to the effect that he now knew it was inconsistent but he hoped it would be useful anyway. (2000-11-01)
Russell's paradox         
In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901.Russell, Bertrand, "Correspondence with Frege}.
D'Alembert's paradox         
  • Jean le Rond d'Alembert (1717-1783)
  • Steady and separated incompressible potential flow around a plate in two dimensions,<ref>Batchelor (2000), p. 499, eq. (6.13.12).</ref> with a constant pressure along the two free streamlines separating from the plate edges.
  • wake]],<br>
•5: post-critical separated flow, with a turbulent boundary layer.
  • Pressure distribution for the flow around a circular cylinder. The dashed blue line is the pressure distribution according to [[potential flow]] theory, resulting in d'Alembert's paradox. The solid blue line is the mean pressure distribution as found in experiments at high [[Reynolds number]]s. The pressure is the radial distance from the cylinder surface; a positive pressure (overpressure) is inside the cylinder, towards the centre, while a negative pressure (underpressure) is drawn outside the cylinder.
  • circular]] cylinder in a uniform onflow.
THE THEOREM THAT, FOR INCOMPRESSIBLE AND INVISCID POTENTIAL FLOW, THE DRAG FORCE IS 0 ON A BODY MOVING WITH CONSTANT VELOCITY RELATIVE TO THE FLUID, IN CONTRADICTION TO REAL LIFE, WHERE VISCOSITY CAUSES SUBSTANTIAL DRAG, ESPECIALLY AT HIGH VELOCITIES
D'Alembert's Paradox; D'Alembert paradox; Hydrodynamic paradox; D'Alembert Paradox; D'Alemberts Paradox; D'Alemberts' Paradox; Dalembert's Paradox; Hydrodynamical paradox; Hydrodynamics paradox; D'alembert's Paradox
In fluid dynamics, d'Alembert's paradox (or the hydrodynamic paradox) is a contradiction reached in 1752 by French mathematician Jean le Rond d'Alembert.Jean le Rond d'Alembert (1752).

Wikipedia

Russell's paradox

In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox published by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains an unrestricted comprehension principle leads to contradictions. The paradox had already been discovered independently in 1899 by the German mathematician Ernst Zermelo. However, Zermelo did not publish the idea, which remained known only to David Hilbert, Edmund Husserl, and other academics at the University of Göttingen. At the end of the 1890s, Georg Cantor – considered the founder of modern set theory – had already realized that his theory would lead to a contradiction, as he told Hilbert and Richard Dedekind by letter.

According to the unrestricted comprehension principle, for any sufficiently well-defined property, there is the set of all and only the objects that have that property. Let R be the set of all sets that are not members of themselves. If R is not a member of itself, then its definition entails that it is a member of itself; yet, if it is a member of itself, then it is not a member of itself, since it is the set of all sets that are not members of themselves. The resulting contradiction is Russell's paradox. In symbols:

Let  R = { x x x } , then  R R R R {\displaystyle {\text{Let }}R=\{x\mid x\not \in x\}{\text{, then }}R\in R\iff R\not \in R}

Russell also showed that a version of the paradox could be derived in the axiomatic system constructed by the German philosopher and mathematician Gottlob Frege, hence undermining Frege's attempt to reduce mathematics to logic and questioning the logicist programme. Two influential ways of avoiding the paradox were both proposed in 1908: Russell's own type theory and the Zermelo set theory. In particular, Zermelo's axioms restricted the unlimited comprehension principle. With the additional contributions of Abraham Fraenkel, Zermelo set theory developed into the now-standard Zermelo–Fraenkel set theory (commonly known as ZFC when including the axiom of choice). The main difference between Russell's and Zermelo's solution to the paradox is that Zermelo modified the axioms of set theory while maintaining a standard logical language, while Russell modified the logical language itself. The language of ZFC, with the help of Thoralf Skolem, turned out to be that of first-order logic.