Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Order types

  1. Dec 3, 2012 #1
    Hi all,

    Went over this today and I'm not grasping it: why is the order type of n + ω = ω, while ω + n ≠ ω? I'd really appreciate if someone could set up the requisite isomorphism in the former. Thanks!
     
  2. jcsd
  3. Dec 3, 2012 #2

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    The set [itex]n+\omega[/itex] is essentially (order-isomorphic to) the following:

    [tex](0,0)<(1,0)<(2,0)<(3,0)<....<(n-1,0)<(0,1)<(1,1)<(2,1)<(3,1)<...<(k,1)<...[/tex]

    Do you see that??

    The isomorphism between the above set and [itex]\omega[/itex] is given by the map T that does the following:

    [tex]T(k,0)=k,~T(k,1)=n+k[/tex]
     
  4. Dec 3, 2012 #3
    Ahh yes, this helps a lot, thanks. So in case of $$\omega + n $$ we could try $$(0,0) < (1,0) < ... < (k,0) < ... < (0,1) < (1,1) < ... < (n-1,1)$$ but we wouldn't be able to set up an isomorphism between this and ##\omega##?
     
  5. Dec 3, 2012 #4

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Yeah exactly. Here we have the natural numbers and we paste n elements after it.
    So take (0,1) for example. That has an infinite number of predecessors. So it can't be [itex]\omega[/itex] since any element in [itex]\omega[/itex] has a finite number of predecessors.
     
  6. Dec 3, 2012 #5
    Excellent, thanks for your help!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Order types
  1. Type of Bayes (Replies: 1)

Loading...