Membership chains

  • Thread starter Dragonfall
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  • #1
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Main Question or Discussion Point

Why do membership chains ([tex]a\in b\in c\in ...[/tex]) have length at most [tex]\omega[/tex]?
 

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  • #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?
 
  • #3
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All chains in [tex]\omega[/tex] are finite.
 

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