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Math Forum / Mathematics / Mathematical Logic / February 2008ZFC inconsistent due to Burali-Forti paradox
The Burali-Forti paradox makes ZFC inconsistent thus proving colin leslie deans claim that mathematics ends in meaninglessness http://en.wikipedia.org/wiki/Burali-Forti_paradox suppose that we associate with each well-ordering an object called its "order type" in an unspecified way (the order types are the ordinal numbers). The "order types" (ordinal numbers) themselves are well-ordered in a natural way, and this well-ordering must have an order type Ω. It is easily shown in naïve set theory (and remains true in ZFC but not in New Foundations) that the order type of all ordinal numbers less than a fixed α is α itself. So the order type of all ordinal numbers less than Ω is Ω itself. But this means that Ω, being the order type of a proper initial segment of the ordinals, is strictly less than the order type of all the ordinals, but the latter is Ω itself by definition. This is absurd!ie a contradiction thus demonstrating deans claim that mathematics ends in meaninglessness -- Message posted using http://www.talkaboutscience.com/group/sci.logic/ More information at http://www.talkaboutscience.com/faq.html elsiemelsi schrieb: > The Burali-Forti paradox makes ZFC inconsistent thus proving colin leslie > deans claim that mathematics ends in meaninglessness [quoted text clipped - 14 lines] > > thus demonstrating deans claim that mathematics ends in meaninglessness You didn't read the wiki page correctly. Not the paradox remains true for ZFC, but a lemma used in the paradox. 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(thanks to http://www.network-science.de/ascii/) You didn't read the wiki page correctly. Not the paradox remains true for ZFC, but a lemma used in the paradox. Did you see the heading: did you read the wiki page fully quote The version of the paradox above is anachronistic, because it presupposes the definition of the ordinals due to von Neumann under which each ordinal is the set of all preceding ordinals, which was not known at the time the paradox was framed by Burali-Forti. Here is an account with fewer presuppositions: [the paradox] remains true in ZFC but not in New Foundations the only way ZFC avoids the paradox is by ad hoc fugging by introducing the Axiom schema of specification Modern axiomatic set theory circumvents this antinomy by simply not allowing construction of sets with unrestricted comprehension terms like "all sets which have property P" -- Message posted using http://www.talkaboutscience.com/group/sci.logic/ More information at http://www.talkaboutscience.com/faq.html elsiemelsi schrieb: > You didn't read the wiki page correctly. Not the > paradox remains true for ZFC, but a lemma used [quoted text clipped - 24 lines] > Message posted using http://www.talkaboutscience.com/group/sci.logic/ > More information at http://www.talkaboutscience.com/faq.html Where does the quote come from? Not from the Wikipage. Note: A quote is supposed to be litteral. Otherwise we call it a paraphrase. To be able to paraphrase one needs intelligence, which elsiemelsi a.shole is obviously lacking. Best Regards you say Where does the quote come from? Not from the Wikipage. Modern axiomatic set theory circumvents this antinomy by simply not allowing construction of sets with unrestricted comprehension terms like "all sets which have property P", as it was for example possible in Gottlob Frege's axiom system. In New Foundations there is a rather different resolution, which is described in that article. and The version of the paradox above is anachronistic, because it presupposes the definition of the ordinals due to von Neumann under which each ordinal is the set of all preceding ordinals, which was not known at the time the paradox was framed by Burali-Forti. Here is an account with fewer presuppositions: suppose that we associate with each well-ordering an object called its "order type" in an unspecified way (the order types are the ordinal numbers). The "order types" (ordinal numbers) themselves are well-ordered in a natural way, and this well-ordering must have an order type Ω. It is easily shown in naïve set theory (and remains true in ZFC but not in New Foundations) that the order type of all ordinal numbers less than a fixed α is α itself. So the order type of all ordinal numbers less than Ω is Ω itself. But this means that Ω, being the order type of a proper initial segment of the ordinals, is strictly less than the order type of all the ordinals, but the latter is Ω itself by definition. This is absurd! -- Message posted using http://www.talkaboutscience.com/group/sci.logic/ More information at http://www.talkaboutscience.com/faq.html |
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