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Membership chains

  1. May 28, 2008 #1
    Why do membership chains ([tex]a\in b\in c\in ...[/tex]) have length at most [tex]\omega[/tex]?
     
  2. jcsd
  3. May 28, 2008 #2

    CRGreathouse

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    They don't. The axiom of infinity gives us a chain [itex]\varnothing\in\{\varnothing\}\in\cdots\in\omega,[/itex], but [itex]\omega\in\{\omega,\cup\omega\}.[/itex] That's a chain of length [itex]\omega+1.[/itex]

    Perhaps you mean [itex]a\ni b\ni c\ni\cdots[/itex], which has length less than [itex]\omega[/itex] by the axiom of regularity/foundation?
     
  4. May 28, 2008 #3
    All chains in [tex]\omega[/tex] are finite.
     
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