In the foundations of mathematics, Russell’s paradox (also known as Russell’s antinomy), discovered by Bertrand Russell in 1901, showed that some attempted formalizations of the naive set theory created by Georg Cantor led to a contradiction. The same paradox had been discovered a year before by Ernst Zermelo but he did not publish the idea, which remained known only to Hilbert, Husserl, and other members of the University of Göttingen.

According to naive set theory, any definable collection is a set. 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 dictates that it must contain itself, and if it contains itself, then it contradicts its own definition as the set of all sets that are not members of themselves. This contradiction is Russell’s paradox.

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The following set of propositions is described by McKenna (2009:1.5)[1] as the classical formulation of the problem of free will:

1)      ‘Some person, at some time, could have acted otherwise than she did.

2)      Actions are events.

3)      Every event has a cause.

4)      If an event is caused, then it is causally determined.

5)      If an event is an act that is causally determined, then the agent of the act could not have acted otherwise than in the way that she did’.

This formulation involves a mutually inconsistent set of propositions, and yet each is consistent with in our contemporary conception of the world, producing an apparent paradox. It is related to another apparent paradox known as Buridan’s Ass.

References

[1] McKenna, Michael, ‘Compatibilism’, The Stanford Encyclopedia of Philosophy(Winter 2009 Edition), Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/archives/win2009/entries/compatibilism/>

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## Zeno’s Achilles and the Tortoise Paradox

Zeno of Elea (ca. 490–430 BCE) was the first person in history to show that the concept of infinity is problematic. In Zeno’s Achilles and the Tortoise Paradox, Achilles races to catch a slower runner–for example, a tortoise that is crawling away from him. The tortoise has a head start, so if Achilles hopes to overtake it, he must run at least to the place where the tortoise presently is, but by the time he arrives there, it will have crawled to a new place, so then Achilles must run to this new place, but the tortoise meanwhile will have crawled on, and so forth. Achilles will never catch the tortoise, says Zeno. Therefore, good reasoning shows that fast runners never can catch slow ones.[1]

Through history, several solutions have been proposed, among the earliest recorded being those of Aristotle and Archimedes. Aristotle (384 BC−322 BC) remarked that as the distance decreases, the time needed to cover those distances also decreases, so that the time needed also becomes increasingly small. Aristotle also distinguished “things infinite in respect of divisibility” (such as a unit of space that can be mentally divided into ever smaller units while remaining spatially the same) from things (or distances) that are infinite in extension (“with respect to their extremities”).[2]

Before 212 BC, Archimedes had developed a method to derive a finite answer for the sum of infinitely many terms that get progressively smaller. Modern calculus achieves the same result, using more rigorous methods. These methods allow the construction of solutions based on the conditions stipulated by Zeno, i.e. the amount of time taken at each step is geometrically decreasing.[3]

[2] Aristotle. Physics 6.9; 6.2, 233a21-31.

[3] Boyer, Carl (1959). The History of the Calculus and Its Conceptual Development. Dover Publications. p. 295.

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Theseus is remembered in Greek mythology as the slayer of the Minotaur. For years, the Athenians had been sending sacrifices to be given to the Minotaur, a half-man, half-bull beast who inhabited the labyrinth of Knossos. One year, Theseus braved the labyrinth, and killed the Minotaur.  The ship in which he returned was long preserved. As parts of the ship needed repair, it was rebuilt plank by plank.

The Ship of Theseus, also known as Theseus’s paradox, is a paradox that raises a question of identity – whether an object which has had all its components replaced remains fundamentally the same object. The paradox is most notably recorded by Plutarch in Life of Theseus from the late 1st century. Plutarch asked whether a ship which was restored by replacing all and every of its wooden parts, remained the same ship.

The paradox had been discussed by more ancient philosophers such as HeraclitusSocrates, and Plato prior to Plutarch’s writings; and more recently by Thomas Hobbes and John Locke. There are several variants, notably “my grandfather’s axe”, and in the UK “Trigger’s Broom”. This thought experiment is “a model for the philosophers”; some say, “it remained the same,” some saying, “it did not remain the same”.[1]

George Washington’s axe (sometimes “my grandfather’s axe”) is the subject of an apocryphal story of unknown origin in which the famous artifact is “still George Washington’s axe” despite having had both its head and handle replaced.

“…as in the case of the owner of George Washington’s axe which has three times had its handle replaced and twice had its head replaced!” [2]

References:

[1] Rea, M., (1995) The Problem of Material Constitution, The Philosophical Review, 104: 525-552.

[2] Ray Broadus Browne, Objects of Special Devotion: Fetishism in Popular Culture, p. 134

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The Grandfather Paradox is one of several metaphysical arguments that attempt to prove that time travel is logically impossible (whether it is physically possible is a question left by philosophers to the physicists).  These arguments all have the same basic form:

Premise 1: If time travel is possible, then X must be possible.

Premise 2: X is not possible.

Conclusion: Therefore, time travel is impossible.

The Grandfather Paradox is described as follows: a time machine is invented enabling a time traveller to go back in time to before his grandfather had fathered offspring.  At that time, the time traveller kills his grandfather, and therefore, one of the time traveller’s parents would never exist and thus the time traveller himself would never exist either.  If he is never born, then he is unable to travel back through time and kill his grandfather, which means he would be born, and so on.

The paradox is also described in this video cartoon.

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