21
edits
Changes
Jump to navigation
Jump to search
Line 1:
Line 1: − [[Bertrand Russell]]'s Paradox revealed a flaw in 1901 in early or naive [[set theory]]. (It is likely others saw this flaw before [[Bertrand Russell]].) It is a form of the [[liar's paradox]] expressed in the terms of [[set theory]].+ − The defect is as follows. Let S be the set of all sets that do not contain themselves as members. In other words, set T is an element of S if, and only if, T is not an element of T: − : <math>S=\{T\mid T\not\in T\}.</math> − Here is the flaw. Is Set S a member of itself? If S is a member of itself, then it cannot be a member of itself by the very definition of S. But if S is not a member of itself, then it must be a member of itself, again by its very definition. Hence there is a fundamental logical contradiction in this type of set, and in any theory that allows it. In formal terms, − :<math> S\in S \iff S\not\in S.</math> − Russel's paradox and others like it can be avoided only at the expense of giving up the idea that any criteria can be used to construct sets. For example in [[Zermelo-Fraenkel]] set theory the construction <math>S=\{T\mid T\not\in T\}</math> would not qualify as [[well-founded]]. Some formulations of set theory would allow such a construction but then S would be a [[proper class]] rather than a set. − ==External Links== − :[http://plato.stanford.edu/entries/russell-paradox/ Russell's Paradox at Stanford Encyclopedia of Philosophy] − [[Category:Set Theory]]+ + + + + + + + + + + + + + + + + + + + + + + + + + +
Replaced content with "IT PROVES THE EXISTANCE OF GOD. ONLY THE CHRISTIAN GOD, MIND YOU."
IT PROVES THE EXISTANCE OF GOD.
ONLY THE CHRISTIAN GOD, MIND YOU.