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Russell's paradox
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Part of the foundations of mathematics, Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russell in 1901, showed that the naive set theory of Frege leads to a contradiction.
It might be assumed that, for any formal criterion, a set exists whose members are those objects (and only those objects) that satisfy the criterion; but this assumption is disproved by a set containing exactly the sets that are not members of themselves. If such a set qualifies as a member of itself, it would contradict its own definition as a set containing sets that are not members of themselves. On the other hand, if such a set is not a member of itself, it would qualify as a member of itself by the same definition. This contradiction is Russell's paradox.
In 1908, two ways of avoiding the paradox were proposed, Russell's type theory and Ernst Zermelo's axiomatic set theory, the first consciously constructed axiomatic set theory. Zermelo's axioms went well beyond Frege's axioms of extensionality and unlimited set abstraction, and evolved into the now-canonical ZFC set theory.
Informal presentationLet us call a set "abnormal" if it is a member of itself, and "normal" otherwise. For example, take the set of all squares. That set is not itself a square, and therefore is not a member of the set of all squares. So it is "normal". On the other hand, if we take the complementary set of all non-squares, that set is itself not a square and so should be one of its own members. It is "abnormal".
Now we consider the set of all normal sets – let us give it a name: R. If R were abnormal, that is, if R were a member of itself, then R would be normal since all its members are. So, R cannot be abnormal, which means R is normal. Further, since every normal set is a member of R, R itself must be a member of R, making R abnormal. Paradoxically, we are led to the contradiction that R is both normal and abnormal.
A longer argument often given for this contradiction, by Russell himself, for example, proceeds by cases. Is R a normal set? If it is normal, then it is a member of R, since R contains all normal sets. But if that is the case, then R contains itself as a member, and therefore is abnormal. On the other hand, if R is abnormal, then it is not a member of R, since R contains only normal sets. But if that is the case, then R does not contain itself as a member, and therefore is normal. Clearly, this is a paradox: if we suppose R is normal we can prove it is abnormal, and if we suppose R is abnormal we can prove it is normal. Hence, R is both normal and abnormal, which is a contradiction.
The paradox holds in intuitionistic logic The formal derivation described above shows that the set leads to a contradiction by showing that assuming true and assuming it false both lead to absurdity; the resulting contradiction implicitly assumes the law of excluded middle. Thus it may be tempting to conclude that the paradox is avoided if the law of excluded middle is disallowed, as with intuitionistic logic. However, as is clear from the first of the two informal arguments presented above, the law of excluded middle is not needed for the paradoxical argument. Here we show that the paradox can still be generated formally by means of the intuitionistically valid law of non-contradiction.
Theorem. The collection is contradictory even if the background logic is intuitionistic.
Proof. From the definition of R, we have that R∈R ↔ ¬(R∈R). Then R∈R → ¬(R∈R). But also R∈R → R∈R (the law of identity), so R∈R → (R∈R ∧ ¬(R∈R)). But by the law of non-contradiction we know that ¬(R∈R ∧ ¬(R∈R)). By modus tollens we conclude ¬(R∈R).
But since R∈R ↔ ¬(R∈R), we also have that ¬(R∈R) → R∈R, and so we also conclude R∈R by modus ponens. Hence we have deduced both R∈R and its negation using only intuitionistically valid methods.
More simply, it is intuitionistically impossible for a proposition to be equivalent to its negation. Assume P ? ¬P. Then P → ¬P. Hence ¬P. Symmetrically, we can derive ¬¬P, using ¬P → P. So we have inferred both ¬P and its negation from our assumption, with no use of excluded middle.
Reciprocation Russell's paradox arises from the supposition that one can meaningfully define a class in terms of any well-defined property ; that is, that we can form the set . When we take , we get Russell's paradox. This is only the simplest of many possible variations of this theme.
For example, if one takes , one gets a similar paradox; there is no set of all with this property. For convenience, let us agree to call a set reciprocated if there is a set with ; then , the set of all non-reciprocated sets, does not exist. If , we would immediately have a contradiction, since is reciprocated (by itself) and so should not belong to . But if , then is reciprocated by some set , so that we have , and then is also a reciprocated set, and so , another contradiction.
Any of the variations of Russell's paradox described above can be reformulated to use this new paradoxical property. For example, the reformulation of the Grelling paradox is as follows. Let us agree to call an adjective "nonreciprocated" if and only if there is no adjective such that both describes and describes . Then one obtains a paradox when one asks if the adjective "nonreciprocated" is itself nonreciprocated.
This can also be extended to longer chains of mutual inclusion. We may call sets a chain of set if for i=1,2,...,n-1. A chain can be infinite (in which case each has an infinite chain). Then we take the set P of all sets which have no infinite chain, from which it follows that P itself has no infinite chain. But then , so in fact P has the infinite chain P,P,P,... which is a contradiction. This is known as Mirimanoff's paradox.
Set-theoretic responsesIn 1908, Ernst Zermelo proposed an axiomatization of set theory that avoided the paradoxes of naive set theory by replacing arbitrary set comprehension with weaker existence axioms, such as his axiom of separation (Aussonderung). Modifications to this axiomatic theory proposed in the 1920s by Abraham Fraenkel, Thoralf Skolem, and by Zermelo himself resulted in the axiomatic set theory called ZFC. This theory became widely accepted once Zermelo's axiom of choice ceased to be controversial, and ZFC has remained the canonical axiomatic set theory down to the present day.
ZFC does not assume that, for every property, there is a set of all things satisfying that property. Rather, it asserts that given any set X, any subset of X definable using first-order logic exists. The object R discussed above cannot be constructed in this fashion, and is therefore not a ZFC set. In some extensions of ZFC, objects like R are called proper classes. ZFC is silent about types, although some argue that Zermelo's axioms tacitly presupposes a background type theory.
Through the work of Zermelo and others, especially John von Neumann, the structure of what some see as the "natural" objects described by ZFC eventually became clear; they are the elements of the von Neumann universe, V, built up from the empty set by transfinitely iterating the power set operation. It is thus now possible again to reason about sets in a non-axiomatic fashion without running afoul of Russell's paradox, namely by reasoning about the elements of V. Whether it is appropriate to think of sets in this way is a point of contention among the rival points of view on the philosophy of mathematics.
Other resolutions to Russell's paradox, more in the spirit of type theory, include the axiomatic set theories New Foundations and Scott-Potter set theory.
Applied versionsThere are some versions of this paradox that are closer to real-life situations and may be easier to understand for non-logicians. For example, the Barber paradox supposes a barber who shaves men if and only if they do not shave themselves. When one thinks about whether the barber should shave himself or not, the paradox begins to emerge.
As another example, consider five lists of encyclopedia entries within the same encyclopedia:
List of articles about people:
| List of articles starting with the letter L:LL!VE TVL&H
...
- List of articles starting with the letter K
- List of articles starting with the letter L
- List of articles starting with the letter M
... | List of articles about places:
| List of articles about Japan:Emperor KoninKatase RiverEnoshima | List of all lists that do not contain themselves:List of articles about JapanList of articles about placesList of articles about people
...
- List of articles starting with the letter K
- List of articles starting with the letter M
...
- List of all lists that do not contain themselves?
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If the "List of all lists that do not contain themselves" contains itself, then it does not belong to itself and should be removed. However, if it does not list itself, then it should be added to itself.
While appealing, these layman's versions of the paradox share a drawback: an easy refutation of the Barber paradox seems to be that such a barber does not exist. The whole point of Russell's paradox is that the answer "such a set does not exist" means the definition of the notion of set within a given theory is unsatisfactory. Note the difference between the statements "such a set does not exist" and "such a set is empty".
A notable exception to the above may be the Grelling-Nelson paradox, in which words and meaning are the elements of the scenario rather than people and hair-cutting. Though it is easy to refute the Barber's paradox by saying that such a barber does not (and cannot) exist, it is impossible to say something similar about a meaningfully defined word.
One way that the paradox has been dramatised is as follows:
Suppose that every public library has to compile a catalog of all its books. The catalog is itself one of the library's books, but while some librarians include it in the catalog for completeness, others leave it out, as being self-evident.
Now imagine that all these catalogs are sent to the national library. Some of them include themselves in their listings, others do not. The national librarian compiles two master catalogs - one of all the catalogs that list themselves, and one of all those which don't.
The question is now, should these catalogs list themselves? The 'Catalog of all catalogs that list themselves' is no problem. If the librarian doesn't include it in its own listing, it is still a true catalog of those catalogs that do include themselves. If he does include it, it remains a true catalog of those that list themselves.
However, just as the librarian cannot go wrong with the first master catalog, he is doomed to fail with the second. When it comes to the 'Catalog of all catalogs that don't list themselves', the librarian cannot include it in its own listing, because then it would belong in the other catalog, that of catalogs that do include themselves. However, if the librarian leaves it out, the catalog is incomplete. Either way, it can never be a true catalog of catalogs that do not list themselves.
Applications and related topicsThe Barber paradox, in addition to leading to a tidier set theory, has been used twice more with great success: Kurt Gödel proved his incompleteness theorem by formalizing the paradox, and Turing proved the undecidability of the Halting problem (and with that the Entscheidungsproblem) by using the same trick.
Related paradoxes
See also
External links
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