1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Ordinals-proof help

  1. Aug 30, 2006 #1
    prove that [tex]\epsilon_0[/tex] is an [tex]\epsilon[/tex] number and that it's the smallest number.
    [tex]\epsilon_0=\lim_{n<\omega}\phi(n)[/tex]
    [tex]\phi(n)=\omega^{\omega^{\omega^{...^{\omega}}}}[/tex] where [tex]\omega[/tex] appears n times.
    an epsilon number is a number which satisfies the equation [tex]\omega^{\epsilon}=\epsilon[/tex].
    for the first part of proving that it's an epsilon number i used the fact that [tex]1+\omega=\omega[/tex], for the second part im not sure i understand how to prove it:
    i mean if we assume there's a number smaller than [tex]\epsilon_0[/tex] that satisfy that it's an epsilon number, then [tex]\omega^{\epsilon^{'}}=\epsilon^{'}<\epsilon_0=\omega^{\epsilon_0}[/tex]
    i know that there exists a unique ordinal such that [tex]\epsilon_0=\epsilon^{'}+\alpha[/tex]
    if [tex]\alpha[/tex] is a finite ordinal then [tex]\epsilon_0=\epsilon^{'}[/tex] and it's a contradiction, but how to prove it when alpha isnt a finite ordinal?
     
    Last edited: Aug 30, 2006
  2. jcsd
  3. Aug 30, 2006 #2

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    I don't know if this is at all valid in the theory of ordinals, but surely lims and exponentials commute, hence e_0 is an epsilon number.

    Secondly, if e is an epsilon number then e=w^e (=>w) = w^w^e (=>w^w) =... hence e must be greater than w, w^w, w^w, w^w^w,.. and therefore e must be greater thanor equal to the smallest ordinal larger than all of w, w^w, w^w^w, etc which is precisely e_0.

    (This is exactly the same as showing that if t is larger than 0.9, 0.99, 0.999,... then t=>1.)
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Ordinals-proof help
  1. Help with proof (Replies: 14)

  2. Proof help (Replies: 6)

Loading...