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Morse-Kelley Class Comprehension axiom and Russell's paradox

  1. Dec 27, 2013 #1


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    As I understand the ZFC solution to Russell's paradox, since {x|x[itex]\notin[/itex]x} must be {x|x[itex]\notin[/itex]x}[itex]\cap[/itex]S for some set S, the paradox goes away, but in Morse-Kelley, if I understand Class Comprehension correctly, although again there must be some M such that {x|x[itex]\notin[/itex]x}[itex]\cap[/itex]M, this M may be a proper class, which no longer is as limiting as the ZFC version, and hence no longer gives the same solution. So either I am going wrong somewhere, or MK solves Russell's Paradox in a different way. I would be grateful for enlightenment. Thanks.
  2. jcsd
  3. Jan 3, 2014 #2
    The Class comprehension Axiom schema states that for any formula phi (with one free variable), there exists a class of all sets satisfying phi. It does not say that there exists a class of all classes satisfying phi. So there's a (proper) class of all sets which do not contain themselves, but there's no class of all classes which do not contain themselves. So Russell's paradox doesn't occur.
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