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Math Help - Immediate successors in Well-ordered sets

  1. #1
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    Immediate successors in Well-ordered sets

    A little help with this question?

    Let be a well-ordered set. Show that for each , either is the greatest element of or has an immediate successor (that is, there exists such that there does not exist with )

    Now, here's how I started:

    Let X be well-ordered by  \leq . Let  x \in X be arbitrary but fixed.

    If x is the greatest element of X, then  \forall y \in X, y \leq x .

    If x is not the greatest element of X, then  x^+ \in X (*) and  x \leq x^+ .

    Then I claim that  x^+ is the immediate successor of x, i.e. there doesn't exist a  y \in X such that  x<y<x^+ (**)

    Questions:

    1. Does the proof depend on the choice of well-ordering?

    2. How do I show (*)?

    3. How do I show (**)?
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  2. #2
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    Quote Originally Posted by Mimi89 View Post
    A little help with this question?

    Let be a well-ordered set. Show that for each , either is the greatest element of or has an immediate successor (that is, there exists such that there does not exist with )

    Now, here's how I started:

    Let X be well-ordered by  \leq . Let  x \in X be arbitrary but fixed.

    If x is the greatest element of X, then  \forall y \in X, y \leq x .

    If x is not the greatest element of X, then  x^+ \in X (*) and  x \leq x^+ .

    Then I claim that  x^+ is the immediate successor of x,


    This isn't usually true


    i.e. there doesn't exist a  y \in X such that  x<y<x^+ (**)

    Questions:

    1. Does the proof depend on the choice of well-ordering?

    2. How do I show (*)?

    3. How do I show (**)?

    Look at the set S:= \{s\in X\;;\;x\leq s\} , and assuming that x isn't the greatest element of X and that X i well-ordered take now a first element in S ...

    Tonio
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  3. #3
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    Thumbs up

    Quote Originally Posted by tonio View Post
    Look at the set S:= \{s\in X\;;\;x\leq s\} , and assuming that x isn't the greatest element of X and that X i well-ordered take now a first element in S ...

    Tonio
    Thank you!!! Though it might seem obvious to you, it didn't occur to me to look at this set and say it must have a minimal element!
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