All horses are not necessarily the same colour as Julian
The horse paradox is the following (invalid) proof of the statement "All horses are the same colour": Image File history File links Some random horses head File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links Some random horses head File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Look up proof in Wiktionary, the free dictionary. ...
We use the principle of mathematical induction. As the basis case, we note that in a set containing a single horse, all horses are clearly the same colour. Now assume the truth of the statement for all sets of at most n horses. Let there be n+1 horses in a set. Remove the first horse to get a set of n horses. By the induction assumption, all horses in this set are the same colour. It remains to show that this colour is the same as that of the horse we removed. But this is easy: put back the first horse, take out a different horse and apply the induction principle to this set of n horses. Thus all horses in any set of n+1 horses are the same colour. By the principle of induction, we have established that all horses are the same colour. Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers, or otherwise is true of all members of an infinite sequence. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ...
The hole in the above "proof" is easy to spot with a little thought: it makes the implicit assumption that the two subsets of horses to which we apply the induction assumption have a common element, but this fails when n = 2. A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. The relationship of one set being a subset of another is called inclusion. ...
Thus this is not a paradox, but merely the result of flawed reasoning; it exposes the pitfalls arising from failure to consider special cases for which a general statement may be false. Robert Boyles self-flowing flask fills itself in this diagram, but perpetual motion machines cannot exist (according to our present understanding of physics). ...
Paradoxes that arise from apparently intelligible uses of language are often of interest to logicians and philosophers.
In fact, Zeno's paradoxes of multiplicity and motion, which revealed problems in the Greek idea of space and time, were resolvable only using mathematics discovered in the 19th century.
Paradoxes which are not based on a hidden error generally happen at the fringes of context or language, and require extending the context or language to lose their paradox quality.
Elevator paradox: Elevators can seem to be mostly going in one direction, as if they were being manufactured in the middle of the building and being disassembled on the roof and basement.
Paradox of hedonism: When one pursues happiness itself, one is miserable; but, when one pursues something else, one achieves happiness.
Supplee's paradox: the buoyancy of a relativistic object (such as a bullet) appears to change when the reference frame is changed from one in which the bullet is at rest to one in which the fluid is at rest
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