The paradox is simply a proof that no village can contain a man who shaves all and only those men in it who do not shave themselves.
Does the barber shave himself? Answer: If the barber shaves himself then he is a man on the island who shaves himself hence he, the barber, does not shave himself. If the barber does not shave himself then he is a man on the island who does not shave himself hence he, the barber, shaves him(self).
If we assume that the barber does not shave himself, we conclude that he must also shave himself; if we assume that the barber does shave himself, we conclude that he cannot; in both cases, a contradiction is engendered. the individual who shaves all and only those individuals who do not shave themselves.
The barber is the "one who shaves all those, and those only, who do not shave themselves". The question is, does the barber shave himself? Any answer to this question results in a contradiction: The barber cannot shave himself, as he only shaves those who do not shave themselves.
Here is the Barber Paradox: In a certain town, the barber shaves all and only those who don't shave themselves. Now, who shaves the barber? If he shaves himself, then he is not one of those who don't shave themselves, so he doesn't shave himself.
Just like the barber who shaves himself, but mustn't, and therefore doesn't, and so must! So now we realise that Russell's Barber's Paradox means that there is a contradiction at the heart of naïve set theory. That is, there is a statement S such that both itself and its negation (not S) are true.
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.
The most commonly discussed form is a contradiction arising in the logic of sets or classes. Some classes (or sets) seem to be members of themselves, while some do not. The class of all classes is itself a class, and so it seems to be in itself. The null or empty class, however, must not be a member of itself.
Russell's paradox (and similar issues) was eventually resolved by an axiomatic set theory called ZFC, after Zermelo, Franekel, and Skolem, which gained widespread acceptance after the axiom of choice was no longer controversial.
The paradox states that if God is all-powerful and all-good, then why is there evil and suffering in the world? This argument has been used to question the existence of God and has challenged theologians and philosophers for centuries.
Classical Logical Paradoxes. The four main paradoxes attributed to Eubulides, who lived in the fourth century BC, were “The Liar,” “The Hooded Man,” “The Heap,” and “The Horned Man” (compare Kneale and Kneale 1962, p114).
In this model, we can only travel back to a particular point in the past only if we had already been to the exact point in our own history. In this model, we can interact with the past, but we cannot alter it. So, the grandfather paradox has zero chance to arise.
The Barber is the one who shaves those and only those who do not shave themselves, and this results in two possibilities: The Barber shaves himself, but the barber cannot shave himself as he only shaves those who do not shave themselves, if he shaves himself, he is no longer the barber.
In 1972 he delivered a budget which was designed to return the Conservative Party to power in an election expected in 1974 or 1975. This budget led to a period known as "The Barber Boom". The measures in the budget led to high inflation and wage demands from Public Sector workers.
Working as a barber must have been an interesting job, to say the least. The red and white stripes of the barber pole originated from a practice known as bloodletting. One of the nastier aspects of barber history, this practice involved drawing blood from the patient, in an attempt to cure them of disease or infection.
The "Paradox"
Applying the definition of the barber to the barber himself, we obtain a contradiction: If he shaves himself, then he must not shave himself. And if he does not shave himself, then he must shave himself.
Number One, Achilles and The Tortoise. How could a humble tortoise beat the legendary Greek hero Achilles in a race? The Greek philosopher, Zeno, liked the challenge and came up with this paradox.
Russell's paradox is often explained simply by using the example of a small town with one barber. The barber shaves all the men from the town who do not shave themselves. If the barber shaves himself, there is a problem because he can only shave men who do not shave themselves.
Bertrand Russell devised what he called the theory of types to prevent the paradox. In this theory, a set would be defined as being of a distinct type, like type 1. The elements of type 1 sets can then only be included in a set of type 2 because sets of type 2 are defined as containing only sets of type 1.
Perhaps the most famous of the paradoxes of infinity, Hilbert's Hotel is a thought experiment where a hotel has an infinite number of rooms, all of which are occupied. Yet, it can still accommodate additional guests, simply by moving them around in a clever way.
In his 'Tractatus logico-philosophicus', Ludwig Wittgenstein declares that he has solved Russell's paradox. He presents the solution in a prima facie simple formula "(∃φ) : F(φu) . φu = Fu". This solution is disregarded both by the Russellians and most Wittgensteinians.
The significance of Russell's paradox is that it demonstrates in a simple and convincing way that one cannot both hold that there is meaningful totality of all sets and also allow an unfettered comprehension principle to construct sets that must then belong to that totality.
C. Roger's (1959) postulation of a self that is separately but causally related to behavior reveals a hidden paradox: If the self is causally related to behavior, how can one respond conditionally to the latter without doing likewise to the former since some element of a cause is always inherent in its effects?
Answer: You cleverly deduce that the first, well-groomed barber couldn't possibly cut his own hair; therefore, he must get his hair cut by the second barber. And, though the second barbershop is filthy, it's because the second barber has so many customers that there's simply no time to clean.