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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:
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:
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:
Hope that helps ...
Peter
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:
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:
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:
Hope that helps ...
Peter
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