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Cantor normal form multiplication

  1. Dec 23, 2009 #1

    daniel_i_l

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    Gold Member

    Let's say that a is an ordinal and it's cantor normal form is:
    [tex] a = {\omega^{\beta_1}}c_1 + {\omega^{\beta_2}}c_2 + ... [/tex]
    I read that
    [tex] a \omega = {\omega^{\beta_1+1}} [/tex]
    But I couldn't find a proof anywhere.
    Can someone give me a source or point me in the right direction so that I can prove it myself?
    Thanks.
     
  2. jcsd
  3. Dec 23, 2009 #2
    The [tex]c_{i}'s[/tex] are positive integers, so [tex]c_{i}\omega=\omega[/tex] and:

    [tex]\left(\omega^{\beta_{i}}c_{i}\right)\omega=\omega^{\beta_{i}}\omega

    = \omega^{\beta_{i+1}}[/tex]

    But the [tex]\beta_{i}[/tex] are in descending order, so their sum is equal to the largest element, which is [tex]\omega^{\beta_{1}+1}[/tex]
     
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