Russell's paradox

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(For E. W. Russell's Paradox, see Religious and militarist attitudes and Paradox supported.)

Part of the foundation 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.

Let M be "the set of all sets that do not contain themselves as members". Formally: A is an element of M if and only if A is not an element of A. In set notation:

$M=\{A\mid A\not\in A\}.$

Nothing in the system of Frege's Grundgesetze der Arithmetik rules out M being a well-defined set. The problem arises when it is considered whether M is an element of itself. If M is an element of M, then according to the definition M is not an element of M. If M is not an element of M, then M has to be an element of M, again by its very definition. The statements "M is an element of M" and "M is not an element of M" cannot both be true, thus the contradiction (but see the section "Independence from excluded middle" below).

Russell's paradox was a primary motivation for the development of higher-complexity set theories with a more elaborate axiomatic basis than simply extensionality and unlimited set abstraction. The paradox drove Russell to develop type theory and Ernst Zermelo to develop an axiomatic set theory which evolved into the now-canonical Zermelo–Fraenkel set theory.

1 Formal derivation
2 History
3 Applied versions
4 Set-theoretic responses
5 Applications and related topics
6 Other related paradoxes
7 See also
8 Footnotes and references
9 External links

[edit] Formal derivation

The following more formal yet elementary derivation^[1] of Russell's paradox makes plain that the paradox requires nothing more than first-order logic with the unrestricted use of the set abstraction. The proof invokes no set theory axioms, and does not implicitly rely on the law of excluded middle. Let Φ(x) be any formula of first order logic in which x is a free variable.

Definition. The collection a, denoted {x : x∈a↔Φ(x)}, is the individual a satisfying ∀x [x∈a ↔ Φ(x)]. All sets are collections, but not conversely.

Theorem. The collection {x : x∉x} is contradictory.

Proof. Replace Φ(x) in the definition of collection by x∉x and obtain ∀x [x∈a ↔ x∉x]. Instantiating x by a yields the contradiction a∈a ↔ a∉a. $\square$

Remark. The force of this argument cannot be evaded by simply deeming x∉x an invalid substitution for Φ(x). In fact, there are denumerably many formulas Φ(x) giving rise to the paradox.^[2]

[edit] History

Exactly when Russell discovered the paradox is not known. It seems to have been May or June 1901, probably as a result of his work on Cantor's theorem that the number of entities in a certain domain is smaller than the number of subclasses of those entities.^[3] He first mentioned the paradox in a 1901 paper in the International Monthly, entitled "Recent work in the philosophy of mathematics." He also mentioned Cantor's proof that there is no greatest cardinal, adding that "the master" had been guilty of a subtle fallacy that he would discuss later. Russell also mentioned the paradox in his Principles of Mathematics (not to be confused with the later Principia Mathematica), calling it "The Contradiction."^[4] Again, he said that he was led to it by analyzing Cantor's "no greatest cardinal" proof.

Famously, Russell wrote to Frege about the paradox in June 1902, just as Frege was preparing the second volume of his Grundgesetze der Arithmetik.^[5] Frege hurriedly wrote an appendix admitting to the paradox, and proposed a solution that was later proved unsatisfactory. In any event, after publishing the second volume of the Grundgesetze, Frege wrote little on mathematical logic and the philosophy of mathematics.

Zermelo, while working on the axiomatic set theory he published in 1908, also noticed the paradox but thought it beneath notice, and so never published anything about it. Zermelo's system avoids the paradox thanks to replacing arbitrary set comprehension with weaker existence axioms, such as his axiom of separation (Aussonderung).

Russell and Alfred North Whitehead wrote the three volumes of Principia Mathematica (PM) hoping to succeed where Frege had failed. They sought to banish the paradoxes of naive set theory by employing a theory of types they devised for this purpose. While they succeeded in grounding arithmetic in a fashion, it is not at all evident that they did so by logic alone. In any event, Kurt Gödel in 1930-31 proved that the logic of much of PM, now known as first order logic, is complete, but that Peano arithmetic is necessarily incomplete if it is consistent. There and then, the logicist program of Frege-PM died.

[edit] Applied versions

There 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: Ptolemy VII of Egypt Hermann Hesse Don Nix Don Knotts Biography of Nikola Tesla Sherlock Holmes Emperor Kōnin	List of articles starting with the letter L: L L!VE TV L&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: Leivonmäki Katase River Enoshima	List of articles about Japan: Emperor Kōnin Katase River Enoshima	List of all lists that do not contain themselves: List of articles about Japan List of articles about places List 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?

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.

[edit] Set-theoretic responses

Russell, together with Alfred North Whitehead, sought to banish the paradox by developing type theory. The culmination of this work, the ponderous and elaborate type theory of Principia Mathematica, does indeed avoid the known paradoxes and allows the derivation of a great deal of mathematics (just how much has never been clearly determined), but it has not been widely accepted.

In 1908, Ernst Zermelo proposed an axiomatization of set theory that avoided Russell's and other related paradoxes. 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 M discussed above cannot be constructed in this fashion, and is therefore not a ZFC set. In some extensions of ZFC, objects like M are called proper classes. ZFC is silent about types, although some contend 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 powerset 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.

[edit] Applications and related topics

The 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.

[edit] Russell-like paradoxes

As illustrated above for the Barber paradox, Russell's paradox is not hard to extend. Take:

A transitive verb <V>, that
can be applied to its substantive form.

Form the sentence:

The <V>er that <V>s all (and only those) who don't <V> themselves,

Sometimes the "all" is replaced by "all <V>ers".

An example would be "paint":

The painter that paints all (and only those) that don't paint themselves.

or "elect"

The elector (representative), that elects all that don't elect themselves.

Paradoxes that fall in this scheme include:

The barber with "shave".
The original Russell's paradox with "contain": The container (Set) that contains all (containers) that don't contain themselves.
The Grelling-Nelson paradox with "describer": The describer (word) that describes all words, that don't describe themselves.
Richard's paradox with "denote": The denoter (number) that denotes all denoters (numbers) that don't denote themselves. (In this paradox, all descriptions of numbers get an assigned number. The term "that denotes all denoters (numbers) that don't denote themselves" is here called Richardian.)

[edit] Reciprocation

Russell's paradox arises from the supposition that one can meaningfully define a class in terms of any well-defined property $P (x)$ ; that is, that we can form the set $P = {x | P (x) is true }$ . When we take $P(x) = x\not\in x$ , we get Russell's paradox. This is only the simplest of many possible variations of this theme.

For example, if one takes $P(x) = \neg(\exists z: x\in z\wedge z\in x)$ , one gets a similar paradox; there is no set $P$ of all $x$ with this property. For convenience, let us agree to call a set $S$ reciprocated if there is a set $T$ with $S\in T\wedge T\in S$ ; then $P$ , the set of all non-reciprocated sets, does not exist. If $P\in P$ , we would immediately have a contradiction, since $P$ is reciprocated (by itself) and so should not belong to $P$ . But if $P\not\in P$ , then $P$ is reciprocated by some set $Q$ , so that we have $P\in Q\wedge Q\in P$ , and then $Q$ is also a reciprocated set, and so $Q\not\in P$ , 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 $P$ "nonreciprocated" if and only if there is no adjective $Q$ such that both $P$ describes $Q$ and $Q$ describes $P$ . 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 $A 1, A 2,..., A n$ a chain of set $A 1$ if $A_{i+1} \in A_i$ for i=1,2,...,n-1. A chain can be infinite (in which case each $A i$ 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 $P \in P$ , so in fact P has the infinite chain P,P,P,... which is a contradiction. This is known as Mirimanoff's paradox.

[edit] Independence from excluded middle

Often, as is done above, the set $M=\{A\mid A\not\in A\}$ is shown to lead to contradiction based upon the law of excluded middle, by showing that absurdity follows from assuming $P$ true and from assuming it false. Thus, it may be tempting to think that the paradox is avoidable by avoiding the law of excluded middle, as with intuitionistic logic. However, the paradox still occurs using the law of non-contradiction:

From the definition of M, we have that M∈M ↔ ¬(M∈M). Then M∈M → ¬(M∈M) (biconditional elimination). But also M∈M → M∈M (the law of identity), so M∈M → (M∈M ∧ ¬(M∈M)). But, the law of non-contradiction tells us ¬(M∈M ∧ ¬(M∈M)). Therefore, by modus tollens, we conclude ¬(M∈M).

But since M∈M ↔ ¬(M∈M), we also have that ¬(M∈M) → M∈M, and so we also conclude M∈M by modus ponens. So using only intuitionistically valid methods we can still deduce both M∈M and its negation.

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.

[edit] Other related paradoxes

The liar paradox and Epimenides paradox, whose origins are ancient.
Curry's paradox (named after Haskell Curry) which does not require negation.
The smallest uninteresting integer paradox.

[edit] See also

Self-reference

[edit] Footnotes and references

^ Adapted from Potter, 2004: 24-25.
^ See Quine, 1938. Incidentally, this theorem and the definition of collection it builds on, are Potter's first theorem and definition, respectively.
^ In modern terminology, the cardinality of a set is strictly less than that of its powerset.
^ Russell, Bertrand (1903). Principles of Mathematics. Cambridge: Cambridge University Press, Chapter X, section 100. ISBN 0-393-31404-9.
^ Russell's letter and Frege's reply are translated in Jean van Heijenoort, 1967.)

Potter, Michael, 2004. Set Theory and its Philosophy. Oxford Univ. Press.
Willard Quine, 1938, "On the theory of types," Journal of Symbolic Logic 3.

[edit] External links

Stanford Encyclopedia of Philosophy: "Russell's Paradox" -- by A. D. Irvine.
Russell's Paradox at cut-the-knot

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Categories: Bertrand Russell | Paradoxes of naive set theory