Proper Subsets of Ordinals ... ... Searcoid, Theorem 1.4.4 .

  • #1
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Main Question or Discussion Point

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 1.4.4 ...

Theorem 1.4.4 reads as follows:



?temp_hash=4bda424fad298c1e9971999ab230fc84.png





In the above proof by Searcoid we read the following:

"... ... Now, for each ##\gamma \in \beta## , we have ##\gamma \in \alpha## by 1.4.2, and the minimality with respect to ##\in## of ##\beta## in ##\alpha \text{\\} x## ensures that ##\gamma \in x##. ... ...


Ca someone please show formally and rigorously that the minimality with respect to ##\in## of ##\beta## in ##\alpha \text{\\} x## ensures that ##\gamma \in x##. ... ...

*** EDIT ***

Is the argument simply that since ##\beta## is the least element of ##\alpha \text{\\} x## ... then if ##\gamma \in \beta## ... then ##\gamma## cannot belong to ##\alpha \text{\\} x## .... otherwise ##\gamma## would be the least element ... and so ##\gamma## must belong to ##x## ...

*** *** ***


Help will be appreciated ...

Peter




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


It may help Physics Forum readers of the above post to have access to the start of Searcoid's section on the ordinals (including Theorem 1.4.2 ... ) ... so I am providing the same ... as follows:



?temp_hash=4bda424fad298c1e9971999ab230fc84.png





It may also help Physics Forum 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:




?temp_hash=4bda424fad298c1e9971999ab230fc84.png

?temp_hash=4bda424fad298c1e9971999ab230fc84.png




Hope that helps ...

Peter
 

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  • #2
andrewkirk
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Is the argument simply that since ##\beta## is the least element of ##\alpha \text{\\} x## ... then if ##\gamma \in \beta## ... then ##\gamma## cannot belong to ##\alpha \text{\\} x## .... otherwise ##\gamma## would be the least element ... and so ##\gamma## must belong to ##x## ...
Almost. Just replace '##\gamma## would be the least element' by '##\gamma## would be less than ##\beta##, so ##\beta## could not be the least element'
 
  • #3
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Thanks Andrew ...

Appreciate your help ...

Peter
 

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