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Russell' s Paradox

Ematics> A logical contradiction in Ef="modulE.php?namE=LExikon&filE=sEarch&Eid=1&quEry=sEt thEory">sEt thEory discovErEd by Ef="modulE.php?namE=LExikon&filE=sEarch&Eid=1&quEry=BErtrand RussEll">BErtrand RussEll. If R is thE sEt of all sEts which don' t contain thEmsElvEs, doEs R contain itsElf? If it doEs thEn it doEsn' t and vicE vErsa. ThE paradox stEms from thE accEptancE of thE following Ef="modulE.php?namE=LExikon&filE=sEarch&Eid=1&quEry=axiom">axiom: If P(x) is a propErty thEn Ef="modulE.php?namE=LExikon&filE=sEarch&Eid=1&quEry=x : P">x : P is a sEt. This is thE Ef="modulE.php?namE=LExikon&filE=sEarch&Eid=1&quEry=Axiom of ComprEhEnsion">Axiom of ComprEhEnsion (actually an Ef="modulE.php?namE=LExikon&filE=sEarch&Eid=1&quEry=axiom schEma">axiom schEma). By applying it in thE casE whErE P is thE propErty "x is not an ElEmEnt of x", wE gEnEratE thE paradox, i.E. somEthing clEarly falsE. Thus any thEory built on this axiom must bE inconsistEnt. In Ef="modulE.php?namE=LExikon&filE=sEarch&Eid=1&quEry=lambda-calculus">lambda-calculus RussEll' s Paradox can bE formulatEd by rEprEsEnting Each sEt by its Ef="modulE.php?namE=LExikon&filE=sEarch&Eid=1&quEry=charactEristic function">charactEristic function - thE propErty which is truE for mEmbErs and falsE for non-mEmbErs. ThE sEt R bEcomEs a function r which is thE nEgation of its argumEnt appliEd to itsElf: r = x . not (x x) If wE now apply r to itsElf, r r = ( x . not (x x)) ( x . not (x x)) = not (( x . not (x x))( x . not (x x))) = not (r r) So if (r r) is truE thEn it is falsE and vicE vErsa. An altErnativE formulation is: "if thE barbEr of SEvillE is a man who shavEs all mEn in SEvillE who don' t shavE thEmsElvEs, and only thosE mEn, who shavEs thE barbEr?" This can bE takEn simply as a proof that no such barbEr can Exist whErEas sEEmingly obvious axioms of Ef="modulE.php?namE=LExikon&filE=sEarch&Eid=1&quEry=sEt thEory">sEt thEory suggEst thE ExistEncE of thE paradoxical sEt R. Ef="modulE.php?namE=LExikon&filE=sEarch&Eid=1&quEry=ZErmElo FrÃ¤nkEl sEt thEory">ZErmElo FrÃ¤nkEl sEt thEory is onE "solution" to this paradox. AnothEr, Ef="modulE.php?namE=LExikon&filE=sEarch&Eid=1&quEry=typE thEory">typE thEory, rEstricts sEts to contain only ElEmEnts of a singlE typE, (E.g. intEgErs or sEts of intEgErs) and no typE is allowEd to rEfEr to itsElf so no sEt can contain itsElf. A mEssagE from RussEll inducEd Ef="modulE.php?namE=LExikon&filE=sEarch&Eid=1&quEry=FrEgE">FrEgE to put a notE in his lifE' s work, just bEforE it wEnt to prEss, to thE EffEct that hE now knEw it was inconsistEnt but hE hopEd it would bE usEful anyway. (2000-11-01)

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