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  1. MHB Master
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    #1
    I am reading Micheal Searcoid's book: "Elements of Abstract Analysis" ... ...

    I am currently focused on understanding Chapter 1: Sets ... and in particular Section 1.4 Ordinals ...

    I need some help in fully understanding Theorem 1.4.3 ...

    Theorem 1.4.3 reads as follows:







    In the above proof by Searcoid we read the following:

    "... ... Then $ \displaystyle \beta \subseteq \alpha$ so that $ \displaystyle \beta$ is also well ordered by membership. ... ...


    To conclude that $ \displaystyle \beta$ is also well ordered by membership, don't we have to show that a subset of an ordinal is well ordered?

    Indeed, how would we demonstrate formally and rigorously that $ \displaystyle \beta$ is also well ordered by membership. ... ... ?



    Help will be appreciated ...

    Peter



    ==========================================================================


    It may help MHB readers of the above post to have access to the start of Searcoid's section on the ordinals ... so I am providing the same ... as follows:








    It may also help MHB readers to have access to Searcoid's definition of a well order ... so I am providing the text of Searcoid's Definition 1.3.10 ... as follows:











    Hope that helps,

    Peter
    Last edited by Peter; October 5th, 2018 at 21:09.

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  3. MHB Master
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    #2 Thread Author
    Quote Originally Posted by Peter View Post
    I am reading Micheal Searcoid's book: "Elements of Abstract Analysis" ... ...

    I am currently focused on understanding Chapter 1: Sets ... and in particular Section 1.4 Ordinals ...

    I need some help in fully understanding Theorem 1.4.3 ...

    Theorem 1.4.3 reads as follows:







    In the above proof by Searcoid we read the following:

    "... ... Then $ \displaystyle \beta \subseteq \alpha$ so that $ \displaystyle \beta$ is also well ordered by membership. ... ...


    To conclude that $ \displaystyle \beta$ is also well ordered by membership, don't we have to show that a subset of an ordinal is well ordered?

    Indeed, how would we demonstrate formally and rigorously that $ \displaystyle \beta$ is also well ordered by membership. ... ... ?



    Help will be appreciated ...

    Peter



    ==========================================================================


    It may help MHB readers of the above post to have access to the start of Searcoid's section on the ordinals ... so I am providing the same ... as follows:








    It may also help MHB readers to have access to Searcoid's definition of a well order ... so I am providing the text of Searcoid's Definition 1.3.10 ... as follows:











    Hope that helps,

    Peter

    I have been reflecting on the above post on the ordinals ...


    Maybe to show that that $ \displaystyle \beta$ is also well ordered by membership, we have to demonstrate that since every subset of $ \displaystyle \alpha$ has a minimum element then every subset of $ \displaystyle \beta$ has a minimum element ... but then that would only be true if every subset of $ \displaystyle \beta$ was also a subset of $ \displaystyle \alpha$ ...

    Is the above chain of thinking going in the right direction ...?

    Still not sure regarding the original question ...

    Peter
    Last edited by Peter; October 4th, 2018 at 22:13.

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