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Encyclopedia > Russell's Paradox

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. The foundations problem in mathematics was the late 19th century and early 20th century term for the search for the simplest metamathematics. ... Bertrand Arthur William Russell, 3rd Earl Russell OM FRS (18 May 1872 – 2 February 1970), was a British philosopher, logician, mathematician and advocate for social reform. ... 1901 (MCMI) was a common year starting on Tuesday (link will display calendar) of the Gregorian calendar (or a common year starting on Wednesday of the 13-day-slower Julian calendar). ... In abstract mathematics, naive set theory[1] was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ... Friedrich Ludwig Gottlob Frege (8 November 1848, Wismar – 26 July 1925, IPA: ) was a German mathematician who became a logician and philosopher. ...


The assumption that sets can be freely defined by any criteria is contradicted by the impossibility of a set containing exactly the sets that are not members of themselves. Such a set qualifies as a member of itself, which then contradicts its own definition (as a set containing sets that are NOT members of themselves).


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: Mathematical notation is used in mathematics, and throughout the physical sciences, engineering, and economics. ...

M={Amid Anotin 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). In mathematics, the term well-defined is used to specify that a certain concept (a function, a property, a relation, etc. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In mathematics, an element (also called a member) is an object contained in a set (or more generally a class). ...


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. In mathematics, this usually refers to some form of the principle, going back to Leibniz, that two mathematical objects are equal if there is no test to distinguish them. ... In set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a set by indicating the properties that its members must satisfy. ... At the broadest level, type theory is the branch of mathematics and logic that first creates a hierarchy of types, then assigns each mathematical (and possibly other) entity to a type. ... Ernst Friedrich Ferdinand Zermelo (July 27, 1871, Berlin, German Empire – May 21, 1953, Freiburg im Breisgau, West Germany) was a German mathematician, whose work has major implications for the foundations of mathematics and hence on philosophy. ... This article or section is in need of attention from an expert on the subject. ... Zermelo–Fraenkel set theory, with the axiom of choice, commonly abbreviated ZFC, is the most common form of axiomatic set theory, and as such is the most common foundation of mathematics. ...

Contents

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. First-order logic (FOL) is a universal language in symbolic science, and is in use everyday by mathematicians, philosophers, linguists, computer scientists and practitioners of artificial intelligence. ... In set theory and its applications to logic, mathematics, and computer science, set-builder notation (or commonly, set notation) is a mathematical notation for describing a set by stating the properties that its members must satisfy. ... This article or section is in need of attention from an expert on the subject. ... The law of excluded middle (tertium non datur in Latin) states that for any proposition P, it is true that (P or ~P). ...

Let Phi(x),! be any predicate of first-order logic in which x,! is a free variable.
Definition. A = {x : x in A leftrightarrow Phi(x)},! is the individual A,! satisfying forall x, x in A leftrightarrow Phi(x),!. All sets are collections, but not conversely.
Theorem. A = {x : x notin x},! is contradictory.
Proof. Replace Phi(x),! in the definition of set with x notin x,! and obtain forall x, x in A leftrightarrow x notin A,!. Instantiating x,! by A,! yields the contradiction A in A leftrightarrow A notin A  blacksquare,!
Remark. The force of this argument cannot be evaded by simply deeming x notin x,! an invalid substitution for Phi(x),!. In fact, there are denumerably many formulae Phi(x),! giving rise to the paradox.[2]

In linguistics and logic, a predicate is an expression that can be true of something. ... In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation for a place or places in an expression, into which some definite substitution may take place, or with respect to which some operation (summation or quantification, to give two... In logic Universal instantiation (UI) is an inference from a truth about each member of a class of individuals to the truth about a particular individual of that class. ...

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. In Zermelo-Fränkel set theory, Cantors theorem states that the power set (set of all subsets) of any set A has a strictly greater cardinality than that of A. Cantors theorem is obvious for finite sets, but surprisingly it holds true for infinite sets as well. ... Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set. ... The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910-1913. ...


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. Mathematical logic is a subfield of mathematics that is concerned with formal systems in relation to the way that they encode intuitive concepts of mathematical objects such as sets and numbers, proofs, and computation. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...


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). Ernst Friedrich Ferdinand Zermelo (July 27, 1871 – May 21, 1953) was a German mathematician and philosopher. ... Zermelo set theory, as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern set theory. ... In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of specification, or axiom schema of separation, or axiom schema of restricted comprehension, is a schema of axioms in Zermelo-Fraenkel set theory. ...


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. Alfred North Whitehead, OM (February 15, 1861 Ramsgate, Kent, England – December 30, 1947 Cambridge, Massachusetts, USA) was an English-born mathematician who became a philosopher. ... In abstract mathematics, naive set theory[1] was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ... At the broadest level, type theory is the branch of mathematics and logic that first creates a hierarchy of types, then assigns each mathematical (and possibly other) entity to a type. ... [...]I dont believe in natural science. ... First-order predicate calculus or first-order logic (FOL) is a theory in symbolic logic that permits the formulation of quantified statements such as there is at least one X such that. ... Gödels completeness theorem is an important theorem in mathematical logic which was first proved by Kurt Gödel in 1929. ... In mathematics, the Peano axioms (or Peano postulates) are a set of first-order axioms proposed by Giuseppe Peano which determine the theory of Peano arithmetic (also known as first-order arithmetic). ... Incomplete is the first single from the Backstreet Boys album Never Gone, released in June of 2005. ... Consistency has three technical meanings: In mathematics and logic, as well as in theoretical physics, it refers to the proposition that a formal theory or a physical theory contains no contradictions. ... Logic (from ancient Greek λόγος (logos), meaning reason) is the study of arguments. ...


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. The Barber paradox is a puzzle attributed to Bertrand Russell. ...


As another example, consider five lists of encyclopedia entries within the same encyclopedia: This article or section includes a list of works cited or a list of external links, but its sources remain unclear because it lacks in-text citations. ...

List of articles about people: List of articles starting with the letter L:

... Ptolemy VII Neos Philopator was an Egyptian king of the Ptolemaic period. ... Hermann Hesse (pronounced ) (2 July 1877 – 9 August 1962) was a German-born poet, novelist, and painter. ... Don Nix was a songwriter, composer, arranger, musician, and author. ... Jesse Donald Knotts (July 21, 1924 – February 24, 2006) was an American comedic actor best known for his portrayal of Barney Fife on the 1960s television sitcom The Andy Griffith Show (a role which earned him five Emmy Awards), and as landlord Ralph Furley on the television sitcom Threes... This article contains additional biographical information about Nikola Tesla that is not on the main page. ... A portrait of Sherlock Holmes by Sidney Paget from the Strand Magazine, 1891 Sherlock Holmes is a fictional detective of the late 19th and early 20th centuries, who first appeared in publication in 1887. ... Emperor Kōnin (光仁天皇 Kōnin Tennō) (November 18, 709[1] – January 11, 782[2]) was the 49th imperial ruler of Japan, according to the traditional order of succession. ... Look up L, l in Wiktionary, the free dictionary. ... Rusty Goffe presents Britains Bounciest Weather, from early 1999 L!VE TV was a British television station that was operated by MGN on cable television from 15 June 1995 - 5 November 1999. ... Wikipedia does not yet have an article with this exact name. ...

  • 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: List of all lists that do not contain themselves:
  • List of articles about Japan
  • List of articles about places
  • List of articles about people

... Leivonmäki is a municipality of Finland. ... The Katase River (片瀬川) is the name of a segment of a river in Shonan, which is in central Japan, about 50 kilometers southwest of Tokyo. ... Utagawa Kuniyoshi, Island Enoshima Enoshima (江の島) is a small island, about 4 km in circumference, at the mouth of the Katase River, which flows into Sagami Bay in Japan. ... Emperor Kōnin (光仁天皇 Kōnin Tennō) (November 18, 709[1] – January 11, 782[2]) was the 49th imperial ruler of Japan, according to the traditional order of succession. ... The Katase River (片瀬川) is the name of a segment of a river in Shonan, which is in central Japan, about 50 kilometers southwest of Tokyo. ... Utagawa Kuniyoshi, Island Enoshima Enoshima (江の島) is a small island, about 4 km in circumference, at the mouth of the Katase River, which flows into Sagami Bay in Japan. ...

  • 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". Look up Layman in Wiktionary, the free dictionary. ... The empty set is the set containing no elements. ...


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. The Grelling-Nelson paradox is a semantic paradox formulated in 1908 by Kurt Grelling and Leonard Nelson and sometimes mistakenly attributed to German philosopher and mathematician Hermann Weyl. ...


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. Alfred North Whitehead, OM (February 15, 1861 Ramsgate, Kent, England – December 30, 1947 Cambridge, Massachusetts, USA) was an English-born mathematician who became a philosopher. ... At the broadest level, type theory is the branch of mathematics and logic that first creates a hierarchy of types, then assigns each mathematical (and possibly other) entity to a type. ... The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910-1913. ...


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. Ernst Friedrich Ferdinand Zermelo (July 27, 1871, Berlin, German Empire – May 21, 1953, Freiburg im Breisgau, West Germany) was a German mathematician, whose work has major implications for the foundations of mathematics and hence on philosophy. ... In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. ... Adolf Abraham Halevi Fraenkel (February 17, 1891 - October 15, German / Israeli mathematician. ... Albert Thoralf Skolem (May 23, 1887 - March 23, 1963) was a Norwegian mathematician. ... The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ... In mathematics, the axiom of choice, or AC, is an axiom of set theory. ... The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ... This article or section is in need of attention from an expert on the subject. ... In foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiom system for set theory designed to yield the same results as Zermelo-Fraenkel set theory, together with the axiom of choice (ZFC), but with only a finite number of axioms, that is without axiom schemas. ... In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. ...


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. John von Neumann (Hungarian Margittai Neumann János Lajos) (born December 28, 1903 in Budapest, Austria-Hungary; died February 8, 1957 in Washington D.C., United States) was a Hungarian-born American mathematician and polymath who made contributions to quantum physics, functional analysis, set theory, topology, economics, computer science, numerical... In axiomatic set theory and related branches of mathematics, the Von Neumann universe, or Von Neumann hierarchy of sets is the class of all sets, divided into a transfinite hierarchy of individual sets. ... The empty set is the set containing no elements. ... Transfinite induction is the proof technique of mathematical induction when applied to (large) well-ordered sets, for instance to sets of ordinals or cardinals, or even to the class of all ordinals. ... In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...


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. At the broadest level, type theory is the branch of mathematics and logic that first creates a hierarchy of types, then assigns each mathematical (and possibly other) entity to a type. ... In mathematical logic, New Foundations (NF) is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types of Principia Mathematica. ... An approach to the foundations of mathematics that is of relatively recent origin, Scott–Potter set theory is a collection of nested axiomatic set theories set out by the philosopher Michael Potter, building on earlier work by the mathematician Dana Scott. ...


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. [...]I dont believe in natural science. ... In mathematical logic, Gödels incompleteness theorems are two celebrated theorems proven by Kurt Gödel in 1931. ... Alan Mathison Turing, OBE (June 23, 1912 – June 7, 1954), was an English mathematician, logician, and cryptographer. ... In computability theory the halting problem is a decision problem which can be informally stated as follows: Given a description of a program and a finite input, decide whether the program finishes running or will run forever, given that input. ... The Entscheidungsproblem (German for decision problem) is the challenge in symbolic logic to find a general algorithm which decides for given first-order statements whether they are universally valid or not. ...


Russell-like paradoxes

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

Form the sentence: A transitive verb is a verb that requires both a subject and one or more objects. ... In grammar, a substantive is either: a noun substantive, now also called simply noun; or a verb substantive, which is a verb like English be when expressing existence (in contrast to use as a copula). ...

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: Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ...

  • 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.)

The Barber paradox is a puzzle attributed to Bertrand Russell. ... The Grelling-Nelson paradox is a semantic paradox formulated in 1908 by Kurt Grelling and Leonard Nelson and sometimes mistakenly attributed to German philosopher and mathematician Hermann Weyl. ... Richards paradox is a fallacious paradox of mathematical mapping first described by the French mathematician Jules Richard in 1905. ...

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) = xnotin 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: xin zwedge zin 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 Sin Twedge Tin S; then P, the set of all non-reciprocated sets, does not exist. If Pin P, we would immediately have a contradiction, since P is reciprocated (by itself) and so should not belong to P. But if Pnotin P, then P is reciprocated by some set Q, so that we have Pin Qwedge Qin P, and then Q is also a reciprocated set, and so Qnotin 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 A1,A2,...,An a chain of set A1 if A_{i+1} in A_i for i=1,2,...,n-1. A chain can be infinite (in which case each Ai 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.


Independence from excluded middle

Often, as is done above, the set M={Amid Anotin 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: The law of excluded middle (tertium non datur in Latin) states that for any proposition P, it is true that (P or ~P). ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Intuitionistic logic, or constructivist logic, is the logic used in mathematical intuitionism and other forms of mathematical constructivism. ... In logic, the law of noncontradiction judges as false any proposition P asserting that both proposition Q and its denial, proposition not-Q, are true at the same time and in the same respect. In the words of Aristotle, One cannot say of something that it is and that it...


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). Biconditional elimination allows one to infer a conditional from a biconditional: if ( A &#8596; B ) is true, then one may infer one direction of the biconditional, either ( A &#8594; B ) or ( B &#8594; A ). For example, if its true that Im breathing if and only if Im... In logic, the law of identity states that A = A. Any reflexive relation upholds the law of identity; when discussing equality, the fact that A is A is a tautology. ... In logic, Modus tollens (Latin for mode that denies) is the formal name for indirect proof or proof by contrapositive (contrapositive inference), often abbreviated to MT. It can also be referred to as denying the consequent, and is a valid form of argument (unlike similarly-named but invalid arguments such...


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. In logic, modus ponens (Latin: mode that affirms; often abbreviated MP) is a valid, simple argument form. ...


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.


Other related paradoxes

In philosophy and logic, the liar paradox encompasses paradoxical statements such as: These statements are paradoxical because there is no way to assign them a consistent truth value. ... The Epimenides paradox is a problem in logic. ... Currys paradox is a paradox that occurs in naive set theory or naive logics, and allows the derivation of an arbitrary sentence from a self-referring sentence and some apparently innocuous logical deduction rules. ... Haskell Brooks Curry (September 12, 1900, Millis, Massachusetts - September 1, 1982, State College, Pennsylvania) was an American mathematician and logician. ... Negation (i. ... The interesting number paradox is a semi-humorous paradox that arises from attempting to classify numbers as interesting or dull. ...

See also

A self-reference occurs when an object refers to itself. ...

Footnotes and references

  1. ^ Adapted from Potter, 2004: 24-25.
  2. ^ See Quine, 1938. Incidentally, this theorem and the definition of collection it builds on, are Potter's first theorem and definition, respectively.
  3. ^ In modern terminology, the cardinality of a set is strictly less than that of its power set.
  4. ^ Russell, Bertrand (1903). Principles of Mathematics. Cambridge: Cambridge University Press, Chapter X, section 100. ISBN 0-393-31404-9. 
  5. ^ 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.

In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ... In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ... Bertrand Arthur William Russell, 3rd Earl Russell OM FRS (18 May 1872 – 2 February 1970), was a British philosopher, logician, mathematician and advocate for social reform. ... Jean van Heijenoort (prounounced highenort) (July 23, 1912, Creil France - March 29, 1986, Mexico City) was a pioneer historian of mathematical logic. ... W. V. Quine Willard Van Orman Quine (June 25, 1908 - December 25, 2000) was one of the most influential American philosophers and logicians of the 20th century. ...

External links


  Results from FactBites:
 
Russell's Paradox (Stanford Encyclopedia of Philosophy) (1417 words)
Russell's paradox is the most famous of the logical or set-theoretical paradoxes.
The paradox arises within naive set theory by considering the set of all sets that are not members of themselves.
Russell's type theory thus appears in two versions: the "simple theory" of 1903 and the "ramified theory" of 1908.
Bertrand Russell - Wikipedia, the free encyclopedia (8395 words)
Russell and Moore strove to eliminate what they saw as meaningless and incoherent assertions in philosophy, and they sought clarity and precision in argument by the use of exact language and by breaking down philosophical propositions into their simplest components.
Russell thought Wittgenstein's elevation of language as the only reality with which philosophy need be concerned was absurd, and he decried his influence and the influence of his followers, especially members of the so-called Oxford school, who he believed were promoting a kind of mysticism.
Russell was an early critic of the official story in the John F. Kennedy assassination; his "16 Questions on the Assassination" from 1964 is still considered a good summary of the apparent inconsistencies in that case.
  More results at FactBites »

 
 

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